EDIT: Sorry for the rewrites – this is almost stream of consciousness. But I think I’ve got it now.
I’m finally back to focusing on this.
So I’ve convinced myself that most conjugations/inflections still create a single 用言 and shouldn’t get any special handling on a clause line. I’ve even given up on the dictionary dot thing (didn’t seem to add much value).
But the て form verbs we’ve been looking at are actually separate clauses. These are compound sentences to my way of thinking.
If I understand correctly, the て examples use something called 連用形 (“continuative form”). It’s discussed in section 2.2 here: 動詞の活用形とその用法をマスターしよう - 国語の文法(口語文法)
The example they give is:
彼に 会って 真相を 確かめる。
I’m going to meet him and find out what’s really going on.
The て in bold indicates where the 連用形 inflection occurred. The plain form verb is, of course, 会う.
The word “continuative” seems to explain why we typically us the word “and” in our English translations (“go and shop and fight and then return and sleep”). It also at least suggests a continuation into (if not dependence on) the final verb when a bunch of 'em are strung together as @kokopelli121123 argued.
It also highlights the fact that Japanese sentences/clauses are quite often just single verbs (English clauses pretty much always require both a subject and a verb).
連用形 (continuative form) verbs can become 連用修飾語 (verb modifiers) per my dictionary:
活用形の一つ。連用修飾語となる。また、文の中止に用いられる(連用中止法)。また、助動詞「た・たい・ます・そうだ(様態)」、助詞「て・ても・たり・ながら」などを付ける。文語では、助動詞「き・けり・つ・ぬ・たり・けむ・たし」、助詞「て・つつ・ながら」などを付ける。
In other words: 会って is the “continuative form” of 会う, but it’s acting as a modifier (連用修飾語) for the verb 確かめる in this sample sentence.
It’s now much clearer to me that each of those て pieces of either example (the fight-a-ninja-and-sleep example or the talk-to-him-and-find-out-what’s-really-going-on example) are well and truly separate clauses. Those are compound sentences.
Here’s how I’d diagram the simpler sentence from the article I linked to above:
彼に 会って 真相を 確かめる。
I’m going to meet him and find out what’s really going on.

The core clause is just 「確かめる」
There are two 修飾語 (modifiers)
-
真相 (the facts; the real situation) which is an object identified with を
-
An entire separate clause which must have the predicate inflected to 連用形 (て form) in order to act as 修飾語. The core of this clause is just 「会って」and it has its own modifier, 「彼に」 indicating who they are meeting with.
-
I thought about “sharing” the subject by extending the vertical line in the top clause down, but it’s possible to add an explicit subject to either or both clauses
彼女が彼に会って私が真相を確かめる (She’s will meet him and I’m going to find out what’s going on)
Each clause has a grammatical subject, but since both are omitted and neither is explicitly stated, it’s safe to assume the same subject. The dotted line indicates both “@” symbols can be assumed to refer to the same person/thing.