An American TV viewer once wrote to a Hollywood trade publication to ask if the hero of the TV science fiction series Quantum Leap, Sam Beckett, was named after the author of Waiting for Godot. After checking with the studio, the paper was able to inform the correspondent that it was sheer coincidence.

It is, though, both deliberate and striking that the central character in Hymn and Cocktail Sticks, two short plays by Alan Bennett that transfer next month from the National theatre to London's West End, has the name "Alan Bennett". It's not the first time that Bennett has written a version of himself on stage: his 1999 adaptation of his short story The Lady in the Van features two characters, both called "Alan Bennett", who represent conflicting responses from the playwright to the plight of Miss Shepherd, a vagrant who is living in a mobile home parked on his drive. In the premiere production, Kevin R McNally and Nicholas Farrell played the two Alans, arguing furiously and, at one point, going to bed together.

Watching the new plays, it struck me that Bennett must have a claim to be the most culturally visible dramatist since Noël Coward; he even made a recent appearance as a special guest star in an episode of the American cartoon Family Guy alongside David Mamet. Paradoxically, though, he is a notably shy man, and has dramatised himself not from self-aggrandisement but from narrative necessity. Both the new double-bill and The Lady in the Van are based on memoirs, and it would be perverse if the "I" of the prose page became, on stage, a Leeds-born writer called say, Andy – or even Sam – Beckett.

Unlike the Italian playwright Dario Fo, who often wrote parts for a version of himself, Bennett has generally only played the eponymous roles by accident; stepping in, for example, when Nicholas Farrell was suddenly summoned to the birth of his child during the London run of The Lady in the Van. In Hymn and Cocktail Sticks, "Alan Bennett" is perfectly captured by Alex Jennings, who offers a person rather than a mere impersonation, although those pretending to be Bennett are certainly helped by the fact that he has an unusually distinctive voice – those lugubrious, self-deprecating West Riding tenor tones – and a wardrobe that approaches a uniform: knotted tie showing in the v of a sweater worn beneath a tweedy greenish jacket.

Anthony Calf, who played a character called Author in David Hare's play The Power of Yes at the National in 2009, had more sartorial freedom. Although implicitly portraying Hare – "Author" is a playwright interrogating witnesses to the global financial crash, as Hare had done while researching the play – Calf wore clothes, including a suede jacket, that would perhaps have been a little less shabby if "Author", like Hare, had the advantage of being married to the fashion designer Nicole Farhi. As with Bennett, Hare's representation of himself was dictated by the drama's structure: unlike earlier verbatim plays, including Hare's The Permanent Way, a study of railway privatisation, the answers given by the masters of the financial universe raised frequent questions, which had to be put by someone.

In most cases, though, dramatists subsume autobiographical material or research experience into a fictional character of the sort that Alan Bennett once described as "the writer in disguise". There is a very Peter Nichols-like playwright in Peter Nichols's A Piece of My Mind (1987), but he is called Ted Forrest. And, in an early draft of Tom Stoppard's Rock 'n' Roll (2006), Stoppard once told me, there was a character called "Tomáš Strausler", the name under which he was born in Czechoslovakia, but the playwright abandoned this strategy as too self-revealing.

There is a separate but related tradition of actor-dramatists who have portrayed on stage fictional characters they have written. Noel Coward originated many of his roles, including Elyot in Private Lives, while Harold Pinter appeared in several of his own plays, including a notable No Man's Land in London in 1992 and an Old Times that toured the US. His widow, Lady Antonia Fraser, also describes in her memoir Must You Go? an evening in the States when Pinter performed an anthology of his characters – and was especially pleased with his own version of Anna in Old Times.

I strongly remember from watching that No Man's Land at the Almeida the special frisson of seeing a writer speaking his own lines – a sensation presumably felt by theatregoers when Coward presented his own texts, although probably not when Shakespeare did (the tradition of the actor-writer being more common at that time). Revealingly, Pinter tended to take his lines much faster than was and is the fashion, pointedly ignoring pauses.

Among current performer-writers, Tracy Letts has left the acting to others in productions of plays such as Killer Joe and August: Osage County, although he continues a separate performing career, most recently in Who's Afraid of Virginia Woolf? on Broadway. The British actor-dramatist Kwame Kwei-Armah has generally kept the two sides of his work apart, too, although he broke the rule for a West End run of Elmina's Kitchen, in which his presence brought a Pinter-like charge.

Kwei-Armah was careful at the time to make clear that he was not trying to show actors how it should be done, and this is also the risk with visible playwrights. Productions of Coward plays were hobbled for years by performers consciously or self-consciously imitating the creator's clipped diction. When, however, Alex Jennings walks on in Hymn, the audience laughs in delighted recognition, knowing at once who he is and has to be. In Bennett's case, the writer in disguise has become the writer undisguisable.