tl;dr; For all intents and purposes, the rumour that John Cleese ad-libbed the name of 'Tim the Enchanter' is completely false.
I found something that purports to be a working copy of the script used in the production of the film, complete with on-the-fly revisions.
This is the 'Monty Python and the Holy Grail' screenplay as it appeared on March 20, 1974. It's a 'working' version of the script., NOT the final script that was filmed. In reading this, you'll be able to see the creative process at work. Many of the scenes were altered from the way they were originally written and others disappeared entirely. Other bits, including some of the funniest and most quoted lines from the film were written into this version of the script, and others were added in later on during filming. I've color-coded the changes so you can follow them more easily.
source: Holy Grail Working Script Part 3
That predates the theatrical release of the film (April 13, 1975) by at least a year. This version of the script has color-coded Cleese's line as BLACK: Regular text. The way the script was originally written.
ARTHUR: By what name are you known?
TIM: There are some who call me Tim?
ARTHUR: Greetings Tim the Enchanter!
unabridged source: Holy Grail Working Script Part 3
If this version of the script's provenance is correct then not only is the line spoken exactly as it later appeared in the released film but the character attribution for the spoken line is Tim.
This evidence coupled with John Cleese's own dismissal of the rumour would seem to prove the rumour false. John Cleese has shown over the years to be a consummate professional and one of the best examples of a straight man in modern comedy. He has exhibited time and again that he can weather the most hilarious situations with a complete deadpan expression while delivering his own lines on cue and per direction. I can see how he might forget a line of dialogue irregularly but would he ever forget which character he was portraying or whose costume and makeup he was wearing?
John Cleese as Tim the Enchanter
I think not. IMHO, John Cleese would never come on set so unprepared.
I have no idea how the rumour got started nor how any internet rumour gets started.
My best guess would be Cleese's comic delivery of the line in question and the subsequent hesitation of the other party. Per the implied stage direction in the script¹, he ends it with a quizzical upnote as if he was asking a question, not stating fact. The comedic effect is putting Arthur and his followers ill at ease; they don't know whether to answer in the affirmative or accept his response. Some fan-boy might have obsessed on the lilted delivery and started a rumour that Cleese was in fact asking who he was supposed to be playing.
¹ The ? at the end of Tim's dialog line (which is notably missing in the OP's version of the script) is the scriptwriters' way of directing the dialogue line to be delivered as a question rather than a statement.