In the first case, you also write non verbal but relevant audio info, but in the 2nd case, this non verbal info can be too much for people who are learning a language, and therefore cannot yet read it very well.
Some DVDs have different tracks for these two cases, whereas online captioning/subtitling platforms - like Amara, DotSUB etc. - don't. Maybe it's better so, otherwise, you'd need to foresee two tracks for each supported language (as online videos come in many languages), and that would be unwieldy.
So for the video where it was not possible to implement these different goals in the same set of English captions/subtitles, which was the starting point of the mentioned off-Amara discussion:
It's not terribly elegant, but it works: should some Briton edit the "English, British" track to make it really "English, British", we'd save their subs for them and revert to the present version, adding an explanatory comment kindly asking them to make do with the existing "English" subs. But that's unlikely, as the video is in American English.
Of course, there are also videos in other languages: for some, there are also separate tracks for local variations that could be used in the same way. And for languages where only one track is foreseen, then one of the multiple tracks of a language that has them can be used: in this case, the explanatory comment had better be done immediately.
Now actually, online subtitling/captioning platforms might perhaps offer another, more elegant solution: allowing users to label their subs as they wish, if none of the options of the drop list fits what they are doing.
I mean something like the possibility offered by YouTube, where you can relabel the subs you are uploading. See e.g. the three relabeled sets of subs in Android 4.0 Accessibility Demo: Turning on Accessibility (framed in yellow in the attached screen shot):
But there's no hurry for that, especially not in Amara, where developers presently have more urgent things to do.