Learn English – Variations of “take care”

This is tricky. Conjunctions and prepositions are among the most difficult meanings for semanticists to describe objectively, and this is why they often have the longest entries in dictionaries.

Note that in the previous sentence, between would not have worked in place of among.

Between implicitly suggests a cline or planes of possibilities - it can be an n-dimensional plane, and whatever it is you're talking about lies somewhere in relation to other, points on it.

The very first sense of 'between' on Collins echoes this sentiment:

1. at a point or in a region intermediate to two other points in space, times, degrees, etc

This is likely why you have intuited that between seems to suggest a star topology in network parlance.

Among(st), on the other hand, is subtly different - it merely suggests a relationship to other things, without making claims to where it stands. If you consider a scatter plot, you could say that any of the points lies amongst others.

There is the possible exception of outliers and those points which lie on the edge of the groups, but that becomes a much more difficult issue to manage, akin to Sorites Paradox. My response to that is that the following sentence seems perfectly grammatical and sensible:

He was the smartest by far among those students in his age group.

The definition for among(st) on Collins suggests a couple of relationships:

• in the midst of
• in the group (of)

Of course, definitions can quickly become circular - that is, how does "in the midst of" differ from "between"? In the midst of seems to invoke a sense of positioning, rather than just that of relationships in general. The positioning can be based on anything, really, but among seems more felicitous when used in the description of vague constellations of meaning and relations, and between in those that are a bit more explicit.

"In the group of", on the other hand, is a much cleaner relationship.

Overall, the difference between between and among seems to be a fuzzy one - in the preceding clause, among would not have worked in place of between. I would say that between is better-suited for more explicit relationships, and among(st) for messier, more vague constellations of relations.

In the case of your two examples, I would say that the first means that you're comparing "[to be] not invited" with other forms. That is, "[to be] not invited" vs form-1, "[to be] not invited" vs form-2, etc.

If the question had read "what are the differences in meaning amongst negated forms (for example [to be] not invited)?", between/among(st) would both work well.

I don't mind usually means that you are allowing for that action to occur. In your case it would be allowing for that person to sit next to you.

I don't care usually mean that you are stubborn, fixed and won't change your mind. This could be associated as being slightly rude/aggressive as for example in an argument:

I don't care about your opinion on my dress because you are a (some discouraging words).

Doesn't matter usually means that the action that is about to occur won't affect the outcome. Also a little bit similar to I don't care in the fixed, stubborn part but it is more polite to use.