All this "BBC News from BBC World" stuff only started when the European Breakfast strand launched in September 2001. Overnight bulletins in the early 2000s were only ever presented as "BBC News". The titles only featured the BBC News logo and the presenters only ever referred to "BBC News". The only reference to BBC News 24 was during the back half hour when BBC World was showing their own recorded programming.
The European Breakfast strand consisted of The World Today at 0500 GMT which was simulcast on BBC News 24 and BBC One and was referred to as "The World Today from BBC News". The 0600 and 0700 GMT bulletins were presented in the same format as the 0500 TWT hour, but were only broadcast on BBC World and were referred to as "BBC News from BBC World" in the opening sequence.
The rest of the news output during the daytime on BBC World (from 0800 GMT until 0100 GMT) was referred to as "BBC World News" on air - and the 2000- titles featured a prominent BBC News logo with a smaller BBC World logo underneath. Going back a bit further, in the late 1990s flags era the titles simply had a "BBC World News" logo on them.
Further confusion was generated in the mid-2000s - as shown in the video clip above - when the new titles had the logos reversed, so BBC World was more prominent than BBC News.
BBC Four's news programme was another naming oddity. When that launched in 2002, it had two different sprinkles that went out on BBC Four and BBC World. The former had a BBC Four News logo for the UK and BBC World's version had a BBC World logo. I've never figured out how they managed to play out two different sprinkles - if anyone can shed any light on that, I'd love to know. Was it simply a case of N9
playing out the BBC Four News version and World Pres overlaying the BBC World version? The post-headlines titles definitely went out from N9 and only featured a BBC logo so that it would be applicable for both channels. To get round this problem, it was renamed "The World" in 2004 and subsequently became World News Today at a later date.
With all that said, it is strange - given how much effort went into keeping things separate and/or generic when simulcasting - that nowadays everything is so sloppy and BBC World News is referred to in the UK overnight.