That's one of my own uploads!

I didn't realise the Trumpet ident was missing online, I've had it recorded all this time!
Did you just have this tape in the VCR all the time and hit record whenever an ident was about to start? Because from that I can guess you went digital sometime mid-1999 (I assume everything from 14:06 is off digital as it's either from Choice/Knowledge or has 'Subtitles' instead of '888' in the corner with the digital-only "in widescreen" continuity announcements).
Yes, you're absolutely right, I would have my finger hovering over the record button during each junction until I had all of the idents recorded.
We first got Sky Digital on 22nd June 1999 according to my diary, so clips recorded from digital do indeed start at that point with the BBC Knowledge ident. Then I got OnDigital in my room on 19th August 1999, the same week it became available in my area. Late in 2000 that I changed the setting to letterbox mode, so then I started recording all the idents again in widescreen, which are on a separate compilation I have uploaded to YouTube.
Is that full clip of the trumpet ident I just get the feeling there something missing like 1-2secs at the start?
I would only start recording if it turned out to be an ident I didn't already have recorded, hence the missing second or two from the beginning of most clips. The Trumpet ident started with the trumpet, daffodil and elephant's trunk in that order, and my recording is missing the first two. I didn't like this particular ident very much so didn't make much effort to try to record it again.
One thing I'm confused about: at 14:50 there's a BBC Choice ident with '888' in the corner. Did digital teletext use 888 for subtitles at one point?
Others have already answered this, but I do recall the 888 caption very early on, however not many BBC Choice or Knowledge programmes were subtitled early on so it was rare to see it.