"All of the み!" 😄
More seriously, all of the ～み here are indeed just the 連用形【れんようけい】 of the verb-forming suffix ～む, with a base underlying meaning of "seems like, looks like, appears like: having that quality". This ～む is very likely cognate with suppositional and volitional verbal suffix ～む, which, via regular sound shifts, became the modern volitional verb ending ～おう (like in 行【い】こう) or ～よう (like in 食【た】べよう). This is also likely cognate with similar suffix ～びる (as in words like 大人【おとな】びる, "to be or seem like an adult"; this had older form ～ぶ, exhibiting not-uncommon //m// ↔ //b// alternation). And, this is all likely also cognate with 見【み】る, and with 目【ま・め】.
(Consider the English cluster around see and seem, look and look like, etc.)
Note that this is looking derivationally, at where things come from and how things have changed through time. My impression from various Japanese-language materials is that most folks may well view ～み as just "a nominalizing suffix that is placed after [形容詞]{けいようし}", and they don't think too much about how it might be related to other things.
(To be fair, most native speakers of any language don't think too much about the language itself -- it's up to us word nerds and language geeks to do that kind of pondering. ✨🥰)
Regarding your observation that some verbs that end in ～む have a 連用形 of ～め～, that's actually a slightly different derivation: あたたむ is a verb, not an adjective + derivational suffix. あたたむ is the older form of modern あたためる. The related adjective + derivational suffix word is instead あたたかむ, which indeed has a 連用形 of あたたかみ.
That said, the ～む that becomes ～める in あたたむ is in turn likely cognate again with the same ～む in あたたかむ -- but that involves a discussion of verb conjugation paradigms that gets quite deep and involved, which would go beyond the scope of this question. :)
