What do Evangelical Protestants mean by "receive Jesus"?

In order to answer, first I think it may be useful to also explain why receiving, accepting, being saved, etc. is important. We as humans are imperfect, and are of sinful nature. Jesus came and died the death that humanity is deserving of so that we might receive salvation and be allowed into heaven for eternity. (See John 3:16) As to the definition of receiving Jesus, to receive Jesus is to admit your sins and ask for forgiveness, believe that Jesus walked the earth and died on the cross and rose again for our sakes, and do the very best you can to emulate Jesus here on earth.

Step 1: Repentance

Romans 3:23

"For all have sinned and fallen short of the glory of God."

Acts 3:19

"Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord."

Proverbs 28:13

"He that covereth his sins shall not prosper, but whoso confesseth and forsaketh them shall have mercy."

Step 2: Believe in Jesus

John 14:1

"Let not your heart be troubled; ye believe in God and also believe in Me."

Romans 10:9

"That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, then thou shalt be saved."

Step 3: Emulate Christ

1 Peter 2:21

"For even hereunto ye were called; because Christ also suffered for us, leaving us an example, that ye should follow His steps."

John 8:12

"Then spake Jesus again unto them, saying, I am the light of the world: he that followeth Me shall not walk in darkness, but shall have the light of life."

John 10:27

"My sheep hear my voice, and I know them, and they follow me."

It's as simple as that.

Firstly, receiving Christ implies being born again in other to commune with Jehovah God, Matthew 16:17 says "And Jesus answered and said unto him, Blessed art thou, Simon Barjona: for flesh and blood hath not revealed{it} unto thee, but my Father which is in heaven".

From the passage above, Jesus disciples were born again so that was why Peter was able to hear from God himself.

Secondly, receiving Jesus into one's life involving being cleansed from all sins, if such is not heeded, it will lead to eternal destruction because,

1 Cor 15:50 says "I declare to you, brothers, that flesh and blood cannot inherit the Kingdom of God, nor does the perishable inherit the imperishable".

So receiving Jesus makes us to be more spiritually alert to the voice of God and less physically alert to voice of the devil. Only those that has received Jesus can inherit the Kingdom of God.

I think you'll come to see that it means different things depending on the context or phase of your maturity as a Christian. Many evangelicals will attest to glorious sounding statements, emphasizing those that the apostle Paul made in his letters, to people they're trying to convert, and gradually begin to frame the topic as a point of great insecurity and tenuous standing. What was the good news for all, "grace by faith alone", is slowly winnowed into the elusive mountain top that consists of a mysterious metric involving works, zeal, to be scrutinized by all professing believers. Throw in a little self-promotion too. You may eventually feel as though you've fallen for a bait-and-switch, but always remember the difference between God's word, and the hearts of man. I recommend you be careful who you trust.

In hopes to give a useful answer I will simply provide two different verses that mention receiving Jesus and quotes from well know evangelical protestants to let them speak for themselves.

KJV John 1:12

But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name.

NIV John 1:12

Yet to all who did receive him, to those who believed in his name, he gave the right to become children of God.

Albert Barnes - Presbyterian

Barnes' commentary on the Bible says "

To as many as received him - The great mass; the people; the scribes and Pharisees rejected him (Jesus). A few in his lifetime received him, and many more after his death. "To receive him," here, means to "believe" on him. This is expressed at the end of the verse."

Colossians 2:6

As ye have therefore received Christ Jesus the Lord, so walk ye in him:

Charles Spurgeon - Baptist

Charles Spurgeon had this to say of Colossians 2:6 in his message Titled "Life and Walk of Faith" delivered on Dec 7th 1862

The idea of receiving, again, seems to imply in it a sense of realization, making the matter a reality. One cannot very well receive a shadow; we receive that which is substantial. Gold, silver, precious stones—such things we can receive; estates, riches, bread, water, food, raiment—all these are things which are substances to us, and therefore it becomes possible for us to receive them. We do not receive a dream; we do not receive, again I say, a shadow; we do not speak of receiving a spectre; we do not receive a phantom. There is something real in a thing that is received. Well now so is it also in the life of faith; we realize Christ. While we are without faith, Christ is a name to us, a person that may have lived a long while ago, so long that his life is only a history to us now! By an act of faith Christ becomes a real person in the consciousness of our heart, as real to us as our own flesh, and blood, and bones, and we speak of him and think of him as we would of our brother, our father, our friend. Our faith gives a substance to the history and idea of Christ, puts real solidity into the spirit and name of Christ, and that which to the worldly man is but a phantom, a thing to hear about, and talk about, becomes to us a thing to taste, and handle, to lay hold upon, and to receive as real and true. I know, ye that are unconverted, that ye think all these things an idle tale; but you that are saved, you who have received Christ, you know that there is substance here, and shadow everywhere else. This has become to you the one grand reality, that God is in Christ reconciling you unto himself.

But receiving means also a third thing, that is getting a grip of it, grasping it. The thing which I receive becomes my own. I may believe it to be real, but that is not receiving it. I may believe, also, that if I ever do get it, it must be given to me, and that I cannot earn it for myself, but still that is not receiving it. Receiving is the bona fide taking into my hand and appropriating to myself as my own property that which is given to me. Now this is what the soul doth when it believes on Christ. Christ becomes my Christ; his blood cleanses my sin, and it is cleansed; his righteousness covers me, and I am clothed with it; his Spirit fills me, and I am made to live by it. He becomes to me as much mine as anything that I can call my own; nay, what I call my own here on earth is not mine; it is only lent to me, and will be taken from me; but Christ is so mine, that neither life, nor death, nor things present, nor things to come, shall ever be able to rob me of him. Oh! I hope, dear friends, you have that blessed appropriating faith which says, "Yes, he is not another man's Christ, he is my Christ," I hope you can look into his face to-day and say, "My beloved, who loved me, and gave himself for me." I hope you do not talk of these things as I might talk of my lord So-and-So's park, and admire its beauties, while I myself have no right to one acre of the many thousands within the park-fence; but I trust, on the other hand, you can say—"The blessings and promises of the Lord my God are all my own; whatever I read of in the covenant of grace that is good, that is comely, that is desirable, I have heard a voice say in my ears, "Lift up now thine eyes, and look to the north, and the south, to the east, and the west—all this have I given thee to be thy possession for ever and ever by a covenant of salt." Now put these three things together, and I think you have the idea of receiving Christ. To receive him is to have him as the result of God's free gift; to realize him; and then to appropriate him to yourselves.

The word "receive" is used in some ten or a dozen senses in holy Scripture; five of them will suffice my purpose just now. To receive is often used for taking. We read of receiving a thousand shekels of silver, and of receiving money, garments, olive-yards, sheep, and oxen. Perhaps in this sense we understand the words of the Master—"No man can receive anything unless it be given him from above," and that other sentence—"To as many as received him, to them gave he power to become the sons of God." We take Christ into us-to return to my old simile-as the empty vessel takes in water from the stream; so we receive Christ. The love, life, merit, nature, and grace of Jesus freely flow into us, as the oil into the widow's vessels.

But the word is also used in Scripture to signify holding that which we take in; indeed, a vessel without a bottom could hardly be said to receive water. I do not suppose any one would talk of a sieve receiving water except in a mock sense. But the life of faith consists in holding within us that which Christ hath put into us, so that Jesus Christ is formed in us the hope of glory. By faith it comes in; by faith it is kept in; faith gives me what I have; keeps what I have; faith makes it mine; faith keeps it mine; faith gets hold of it with one hand, and then clasps it with both hands with a grasp that neither death nor life can loose. Then, receiving sometimes means in Scripture simply believing. "He came unto his own and his own received him not." We read of receiving false prophets, that is, believing them. Now, to receive Christ is to believe him. He says, "I can save you."—I receive that. He says, "I will save you."—I receive that. He says, "Trust me and I will make you like myself."—I receive that.

Whatever Jesus says, I believe him, and receive him as true. I make his word so true to myself that I act upon it as being true, and regard it not as a word that may possibly be true, but which must be true, even if heaven and earth should pass away. This is receiving Christ—believing what he has said. Receiving, also, often signifies in Scripture entertaining. Thus the barbarous people at Melita received Paul and his companions kindly, and kindled a fire. Ah! after we have once found all in Christ to be our own, and have received him into ourselves by faith, then we entreat the Lord to enter our hearts and sup with us. We give him the best seat at the table of our souls; we would feast him on the richest dainties of our choicest love. We ask him to abide with us from morn till eve; we would commune with him every day, and every hour of the day. We entertain him; we have a reception-chamber in our hearts, and we receive Christ. And then, once again, receiving in Scripture often signifies to enjoy. We hear of receiving a crown of life which fadeth not away; that is, enjoying it, enjoying heaven, and being satisfied with all its bliss. Now, dear friends, when we receive Christ, there is intended in this an enjoying of it. I am only now talking the simplicities of our faith, but I do want to make them very personal to you. Are you thus enjoying Christ? if you had a crown you would wear it; you have a Christ—feed on him. If you were hungry and there were bread on the table, you would eat. Oh! eat and drink, beloved, of your Lord Jesus Christ. If you have a friend, you enjoy his company: you have a friend in Christ; Oh! enjoy his conversation. Do not leave him, like a bottle of cordial for the fainting, sealed up from us; let him not be as some choice dainty all untasted, while you are hungry. Oh! receive Christ, for this is the very heaven and rest of the soul. His flesh is meat indeed, his blood is drink indeed. Never did angels taste such divine fare. Come hither saints and satiate yourselves in him. To take him into one's self, to hold him there, to believe every word he says, to entertain him in our hearts, and to enjoy the luscious sweetness which he must confer upon all those who have eaten his flesh, and have been made to drink of his blood—this it is to receive Christ.

But we have not brought out the real meaning of this life of faith yet till we dwell upon another word. As ye have received. Received what? Salvation may be described as the blind receiving sight, the deaf receiving hearing, the dead receiving life; but beloved, beloved, here is a thought here—oh that you may get hold of it! We have not only received these things, but we have received CHRIST. "As ye have received Christ Jesus the Lord." Do you catch it? It is true that He gave us life from the dead? He gave us pardon of sin; He gave us imputed righteousness. These are all precious things, but you see we are not content with them; we have received Christ himself. The Son of God has been poured out into us, and we have received him, and appropriated him. Mark, I say, not merely the blessings of the covenant, but himself; not merely the purchase of his blood, but he himself from whose veins the blood hath flowed has become ours; and every soul that hath eternal life is this day a possessor of Christ Jesus the Lord.