I would never try to convince a Dutch speaker of anything by mere argument, but I'll make some responses for you to mull over at your leisure.
1. Why do the Japanese pros bother to distinguish these terms?
2. (Expanding on what I said above) A jimoyo is literally a territorial framework. A moyo is just a framework. A jimyo is one where "an area territory cannot be said to be established yet, but
it is not a mere moyo; the probability of it turning into territory is rather high" (quoting from a Japanese go dictionary; my emphasis).
3.
Then there's Takemiya's kind of moyo
Agreed he was special, but it's the exception that proves the rule. If you want to keep him in the framework (so to speak) covered by the ordinary rules, you can also consider his styles in other ways. The obvious one is that he made omoyos (big-scale moyos) which have the property that they can fission off and form multiple moyos of (eventually) differing types. You can also turn this around and, based on the fact that Japanese normally doesn't distinguish singular and plural, say that we should talk about Takemiya moyos (plural) that just happen to be adjacent. Matthew Macfadyen's concept of virtual territory is especially useful in that regard.
4. I'm not disputing that one and the same position can, pending the next decisive move,
superficially be interpreted as a territorial framework, a framework or a sphere of influence. But they are not synonymous. The choice of term depends on the strategic use to which the position is put. The strategy comes first, then the term.
5. There's a similar issue implied in your response:
In the past I would have used "invade" for such a deeper entry but since studying with KataGo I have understood that real effective invasions not merely try living in the opponent's sphere of influence but rather dislodge it altogether and aim at the weaknesses of the surrounding positions. Friendly stones can help in that regard, which is the aji they provide.
To which I want to say, "Precisely." Western amateurs love to invade. They think of the tasty word and then are led by the nose by it. Somewhat stronger amateurs have learned to keep two words in mind: "invade" and "reduce." So, sometimes they invade, and sometimes they reduce. But in each case the word is still leading them by the nose. But take the word used in old Chinese: 侵 (qin). It can be used of an invasion, a reduction, a peep or an incursion along the edge, and other things. But (I postulate) in no case did Huang Longshi or the like ever think of any of these words specifically. At best he would thing of an overarching term: namely qin, which encompasses all these possibilities. The way I render it to use the word 'encroach'. Apart from the fact that it covers all the above possibilities, it is a rarish word in English and so does not have too many distracting associations.
Now if you look at the starting position above and, instead of saying yo yourself, "how do I invade" or "how do I reduce" or even "how do I play on the sector line", you say "how do I encroach," I think you will evolve a very different and richer mindset. The implication of your quote is that you have already done that to an extent. It's just that, instead of saying "how do I encroach", you are saying "what does katago do here". One of the things you've observed is that friendly stones can be a big help in these situations. Ideas stemming from encroaching will show exactly the same sort of thing. The beauty of using encroach as your high-level starter word, is that you don't even need to worry about whether an area id a framework or a sphere or a gefurtel. (Similarly, incidentally, if Black above thinks in higher-level terms of how do I stop the opponent encroaching, as distinct from how do I stop him from invading or reducing, he will end up with a richer palette of ideas.)
You can go further along the same route. You can ask yourself, how does katago's choice help my other groups in the other, remote areas of the board (and you will find that it does indeed so that). Alternatively, you can use the same concept dressed up in human terms: zhaoying (call & response). Either way, the idea is to think first at a higher strategic level. Strategy first, term second. Otherwise you end up with a 'special operation.'
that is to say, it features among its candidate moves and doesn't drop too many points with respect to the "blue move". It pictures 
and 
then Black plays elsewhere. In this no komi game White is just a little behind. The reduction has happened on the sector line but we need to also note how