[Gary Baldy]

You already listed some of the reasons, but here's what I think:

Anyone who's won too many shows before
Anyone with a criminal record
Anyone who's already been on the same show, unless they're a celebrity
Anyone who's a celebrity, unless it's a celebrity edition of the show
Anyone who is an employee of the production company or broadcaster, and their families
Anyone proven to have cheated when previously on a show

How do they define celebrity though?

Rob Curling was on public Eggheads once.
Henry Kelly, famous for roughly the same sort of thing, was on Celebrity Pointless.

Are “reality stars” celebrities? I recall Jade Goody being on public Eggheads once too- a lot of the ITV2/BBC3/E4 shows great reality stars as celebrity. But then Ashleigh from BB15 was on Lingo as a member of the public. Zoe Burkett from Pop Idol 1 was on Big Brother 15 as a member of the public. A few months back, a former Apprentice contestant appeared on Countdown as a member of the public, but then you have people like Thomas Skinner doing the celebrity circuit even though his claim to fame is still “being a contestant on the apprentice”.

Frederick Forsythe the author was on public Eggheads.

Actress Emma Williams was on public Beat the Chasers, even though all the chasers recognised her as a celebrity.

There was a woman from Honeyz on The Voice as a member of the public, Kevin Simm from Liberty X also won it as a member of the public.

Sophie Evans, West End star, appeared on a celebrity edition of Pointless and then appeared on public BGT a few years later. Loren Allred sang Never Enough on the Greatest Showman, one of the biggest films of all time, and appeared on BGT as a member of the public.

A few years ago there was an actress from Eastenders whose name escapes me who took a supermarket job because she was skint and out of work- so the whole “we can’t have celebrities on because they’re already rich” doesn’t always really work- if the actress in question went on the public version, would people be saying “why isn’t she on the celebrity version?”

So who determines where “member of the public” ends and where “celebrity” starts?