Here's my unofficial and hopefully concise history:
From 1986 to 1991, BBC2 used this ident ('symbol' in BBC parlance). It was mostly static and had no soundtrack.
Sometime around (I'm guessing) 1989 or 1990, the BBC commissioned the design and branding agency Lambie-Nairn to revamp BBC1 and BBC2. (They previously had made these Nine O'Clock News titles
As part of audience research, it was discovered that BBC2 had a stuffy, pretentious image with most viewers. So BBC2's rebrand was focused on getting rid of that association, hence the idents being very different: much more animation/motion, using real models instead of computer graphics, each ident having their own soundtrack, and so on.
It worked and the viewer research discovered that the perception of the channel had become very positive ('sophisticated, witty, amusing'), without any of the actual programmes changing.
Most of this has been taken from this segment from the programme
How Do They Do That?
which I recommend watching:
Another TV Forumer, Jonny, has previously posited that the point at which the '2's really took off was in 1993 when a new set of idents were introduced:
Worth noting the famous 2s themselves weren't initially met with universal acclaim and it was arguably the second batch in 1993, with the more animated 2s like Car and Dog, that really built an emotional connection with the wider public (and subsequent admiration for the identity as a whole).
BeanosOnToast and fanoftv gave kudos