Check here what admission class you got (WT or J1). If it says J1, don’t worry – the electronic record is superior to the passport stamp.
If it says WT, however, you must get it fixed immediately by going to a deferred inspection site, a list of which can be found here. Otherwise, as soon as you start performing your J1 activity, you’ve broken immigration law and will be deported if caught.
UPDATE: Given the reply you got:
The class of admission and admit until date indicated on your electronic I-94 record is correct.
You presented yourself as a Visa Waiver Program applicant and utilized the Automated Passport Control kiosk on your last entry into the United States.
You were admitted in WB status (unsurprisignly, because the electronic record invariably contains the correct info) because you probably presented your passport open at the main ID page (after using the kiosk, which is not for J1 visa holders) and didn’t present your DS-2019.
WB/WT status cannot be converted to any other status
Unless the CBP make an exception and agree to correct it, you’ll have to leave the US (Canada and Mexico are fine), and re-enter, asking to be admitted in J1 status. Present the passport open at the visa page as well as your DS-2019.
Bear in mind, if entering from Canada or Mexico by land, a white form will be stapled into your passport (ask immigration to create an electronic I-94 as well in case you lose the white form – at major crossings they’re usually able to). Keep this during your whole stay in the US, and hand it in at the check-in desk when it’s time to go home.
It is important what status you were admitted in, regardless of what valid visas you have in your passport. This becomes vital IF you are apprehended for some reason or accused of violating status or maybe need to extend duration of stay.
Being admitted under VWP gives you fewer immigration rights if something happens. You forfeit the right to an immigration judge if something happens and they decide to deport you. Also it is annotated in your immigration record.
If you were admitted incorrectly to the United States, you should
visit a local Customs and Border Protection (CBP) Deferred Inspection
Site or Port of Entry (POE) that has a Deferred Inspection office to
have his or her admission corrected. See list of Deferred Inspection
Site locations and list of Port of Entry locations.
5 Oct, 2023
12 Sep, 2023
8 Sep, 2023
18 Aug, 2023
18 Aug, 2023
18 Aug, 2023
13 Aug, 2023