PS Going back to DA's original question, about whether there's a rule, I haven't met one and am not aware of one. The writer of that article mentions this (rather shrilly ) towards the end of the article:
The 'simple' (that is, hard for students) fact is that some words have it ('button', 'mutton'...) and some words don't (as in the placename 'Luton' - in which there's a schwa [preceded, sometimes, by a glottal stop]). It's possible that it varies from speaker to speaker; for example, I don't do it with 'latin', but I gather from DA's post that some people do. I'll think further about this....The insidiousness of it is that the native speaker cannot really pinpoint for the non-native what the problem is except to say unhelpful things like "try harder" or "listen carefully," followed by "no, that's not it. no, that's not it. yeah, that's it! no, that's not it - say it like you said it before! no, that's not it." ...
Retired English Teacher