> “There’s” is a contraction for either “there is” or “there are”

But doesn't con-traction mean "pulling together"? You pull the last letter over towards the first ones, squishing out the ones between. Only there is no 's' at the end of "there are" to pull over next to "there".

So "there's" can't really be a contraction of "there are", AFAICS.

> Written contractions are meant to faithfully represent spoken English, in which people indeed say “there’s” for both the singular and the plural.

Sure, it may be a perfectly valid usage, so it's something... But as matter of terminology, whatever that something is, I don't think it's a contraction.

The way I've explained the correctness of the plural use to a non-native speaker, is that (to use the cats example) there is one group of cats, the group is singular even though its content is plural, so “there is” is correct, therefore so is the contraction. Not explicitly mentioning the group is just another abbreviation, because it is implicitly understood. This avoids the discussion of whether “there are → there's” is valid in written form (which tends to be more strict than verbal use).