@Zizka There you go. I was almost done with the analysis when you posted your message.
Everyone: I hope my analysis helps you understand how I think about the て. I won’t guarantee that it works every time. To be honest, I think how my brain works with language is that, once I feel like I’ve managed to get a multi-purpose ‘gist’ out of everything I’ve seen about a structure, I will look for every possible way to twist and massage that gist to fit into every single function of that structure. I look for every possible connection I can find. Maybe it’s ‘lateral thinking’, I don’t know, but if there’s a way to slightly warp my understanding of the ‘core meaning’/‘gist’ of a structure that allows it to fit what it specifically means in context, I will do it.
I don’t have the textbook that I first used to learn Japanese on me, so I can’t check how the て-form was first explained to me. @YanagiPablo might have a copy, and I guess he can look inside if he’s interested. After a lot of searching, I found a few old photos of the book that I took while doing the lesson translations, and all I can tell you is that, in the literal translation, all the て-forms are written using the infinitive form in French, which is exactly the same as how they listed the present tense verbs at the end of sentences. It’s really just meant to be a way to list actions that occur together. (I mean that in the broad sense of ‘happening around the same time within the same context’).