Do British peers have a special way of signing their names?

From an Australian biographical dictionary:

On 9 February it was announced in the press that Forrest had been recommended for a barony, the first native-born to be so honoured. He was delighted, and thereafter signed only his surname, but was more than a little aggrieved that he was not also prime minister. _{F. K. Crowley: "Forrest, Sir John (1847–1918)"}

Which leads us to look for a pattern, that Cracroft's Peerage –– The Complete Guide to the British Peerage & Baronetage is happy to answer:

Even foreign place names are chosen in, for instance, cases where soldiers and sailors who have been ennobled have rendered distinguished service abroad, though it is usual to associate with the title home place names also. Thus Lord Nelson's first peerage was "BARON NELSON, of the Nile and of Burnham Thorpe in the County of Norfolk", and when, after his death, his honours went to his brother and an earldom was added, the earldom was "of Trafalgar and of Merton in the County of Surrey". The Duke of Wellington was "VISCOUNT WELLINGTON, of Talavero and Wellington in the County of Somerset". In modern times Sir David Beatty was created "BARON BEATTY, of the North Sea and of Brooksby in the County of Leicester", and Field-Marshal Byng was raised to the peerage as "BARON BYNG OF VIMY, of Thorpe-le-Soken in the County of Essex".

So that, as I have said, "BARON BLANK, of Blanktown in the County of Blankshire", is the essential form in which a new title is set out in the Letters Patent in the normal case. This is the form in which the announcement of his peerage subsequently appears in The London Gazette, but it is not his title. That is simply "Lord Blank", and his signature will be merely "Blank". It is not correct for him to be referred to as, or for him to sign himself, "Blank of Blanktown".

Nevertheless there has been an increasing tendency of recent years for newly-made peers to adopt this style and to try to convince others of their right to do so. That deservedly popular and philanthropic peer, the late Viscount Wakefield, invariably signed himself "Wakefield of Hythe", both as baron and viscount, and most writers and newspapers followed his example and referred to him accordingly. But there is nothing in the record of the honours bestowed on him to justify this departure from the normal.

Not long ago it was stated in a popular commentary in one of our most reputable newspapers:

Except where it is necessary to differentiate a title from another of the same name the "of etc." should not be used. I mention this because this rule is sometimes broken by newly-made peers, and often by journalists.

On formal documents, however, the full title is used. This is why Lord Tyrrell's signature, which is thrown on the screen as Film Censor, is Tyrrell of Avon.

With all respect to the writer of that comment his own paragraph was about as full of errors as it could be. He was wrong when in an introductory sentence which I need not quote he referred to "Lord Cherwell of Oxford", and he was wrong in his final "explanation" of when the "of etc." should be used.

A peer's signature is the same whether it is on a formal or an informal document, and, again with all respect to an eminent public servant, Lord Tyrrell was wrong when he appended the signature "Tyrrell of Avon" - whether it be to a film or any other document.

The rules on the point are quite clear. They have been understood and observed for a very long time, but they were definitely formulated by King George V under the guidance of a former Garter Principal King of Arms, the late Sir Henry Burke.

There is some variation throughout the British Isles and times. Fashion doesn't stop at clothes.

A peer's surname was his title. He was Devonshire and not Cavendish, the family name. The children used the family name, he went by the title. A peer's signature was his title .. Wellington, Jersey, Rutland, Norwich. They did not use their surnames. They generally did not introduce themselves as "John Johnson, Earl of Marsh," but as "Marsh." He would sign dispatches, letters and other things with just his title- Marsh. His wife would use his title as a surname and sign as E. Melbourne , or Elizabeth Melbourne. Lady Melbourne even sent a letter or two signed with just a Melbourne as her husband did. I was told that this was correct but most of the examples I have seen have the woman signing with her first name or initial and the title.
Do not mix courtesy and peerage titles. If a man is a peer he is never Lord First name anything.
_{Nancy Mayer, A most proper authority on all things Regency: "Speaking To And Of Titled Persons"}

In contrast to:

Not every knight is a lord; not every lord is a knight. It is best not to say My Lord to anyone not so entitled.
A territorial title is one which is attached to a particular piece of land, such as a county.
Peers sign their names and refer to themselves and each other by their territorial titles, such as "Henry Southampton", "Francis Bedford", or "Thomas Rutland".
_{Life in Elizabethan England: Titles and Forms of Address}

In the old days (meaning pre-Tudor times) the land of Britain was divided into parts each assigned to a man bound to the king by oaths of fealty and often blood ties in one way or another. Such men were responsible for supplying the king with soldiers, if need be, and themselves serving as knights. These men were generally known by the name of the land to which they were assigned. So, for example, Ralph Neville was created Earl of Westmorland by Richard II in 1397. Since this fiefdom was the greatest of all his possessions at the time, he would be known as "Westmorland" in certain contexts, especially military ones.

There are an important reasons for this involving honor and respect. For example, let's imagine I am speaking to the king about Neville's possible participation in a war, I will not call him "Robert Neville", because that would imply I know him, to use his name in a familiar way. I might be construed as implying I were his friend, and even if I were his friend, it would be rude of me to throw his personal name out in front of the king and others. This might get back to him, people saying "Durden was calling you Neville to the king". If I were not his close friend, he might be very insulted by this. It is like those low people and magazines nowadays who refer to Hollywood celebrities by their first names, as though they were somehow their friends.

To show proper respect to such people I would refer to them by their fief or by their title.

Later on, such people became private persons, but the same customs of nomenclature and respect continued, so that titled people would even refer to themselves by their baronage, no matter how small it was, as to suggest their importance.

Now, of course, such usages are quite conceited, because even Ralph Neville, a real peer, would always sign his personal name "Ralph Neville" to any document or letter. In other words, other people would refer to him as Westmorland, but he himself would use his own name. So, when someone like John Forrest starts signing his name as "Forrest" he is putting on airs as though he is some great man, without really understanding the custom in the first place.

Legal Note

I should probably point out that there is point of legality that is significant here. At a certain point the British parliament made laws respecting the use of one's name, and a special exception was created for peers (meaning a member of the House of Lords) whereby they could legally sign their name by the name of their estate, thus "Marlborough" or "Norfolk" or whatever. Over the years, some people considered it significant whether you had the "legal" right to sign your name in this fashion. In general, it was the custom only for those with inherited titles from landed estates to use the title itself. Those, like Forrest, who received their lordship from a civil source would use their surname. For example, Frederick Stanley was known as "Lord Stanley", not "Lord Derby".