Note: I wasn't sure whether to post this as an answer, or as its own question. I mostly read philosophy of ethics and this is a little outside my usual area. Please be ruthless!

That doesn't read badly at all, and I thought it was a really good statement of the compatibilist case, and it's how I would argue it as a compatibilist (actually, I'd probably get more confused than you and make more mistakes, but never mind).

A compatibilist probably has a conception of a person having a true essence of some kind distinguishable from the Universe around it. I take this stance from Hume leading in to Schopenhauer, who are the earliest philosophers with a coherent compatibilist stance who I personally am aware of - and I know Schopenhauer took a lot from Hume simply because Schopenhauer often says how much he loves Hume. And I use the earliest compatibilist stance I can find because I think these are the simplest and easiest formulations to understand for me and my argument here.

So with this true essence, although our true essence is itself caused by the Universe, it is distinguishable from the Universe - i.e. we could pick up a person and plop them in a different Universe, and this concept makes sense and the person is able to act in a way we can recognise as them, not the old Universe or the new Universe. The task for the hard determinist is then to show either that the concept doesn't make sense, or that the concept isn't meaningful, i.e. either dropping 'a person' in a different Universe is impossible (the act of dropping perhaps blurs the boundaries between Universes in such a way that makes them both part of the same Universe, or a person is so tightly bound up with the Universe that a different Universe changes the essence of that person, or there's no such thing as 'a person' in a way that makes sense for free will), or that changing their Universe doesn't change anything meaningful.

The way randomness could introduce libertarian free will to a causal universe seems to be that it could provide a truer true essence to people, i.e. an essence that in some way stems from an 'arbitrary' seed, and this arbitrary seed - now utterly and completely distinct from the Universe at large - represents the beginnings, the tools, the foundation, on which your personality and thus your agency and your free will, is built. You could say that it is only the arbitrary seeds of your 'self' that represent your 'true self', and thus only these arbitrary seeds that represent your soul.

Also, with a random seed somewhere in your, uh, 'soul', an observer cannot perfectly predict your actions. This moves free will from the noumenal realm, where from the inside it is obvious that I can act and affect things, to the phenomenal realm, where it is a bit less obvious that I can change my 'fate' because people can observe my motivations and say things like "he did that because he was angry, because he watched Zoolander 2; his emotions controlled him, and were in turn controlled by a bad movie". With a random seed a phenomenal free will is also possible - we can rationalise my actions after the fact, but they're not predictable in advance even if you know everything about the Universe that exists outside the bounds that make me a person.

I also think that I can be usefully confused by the idea of systems that are wholly unpredictable but mechanical. I think that we can tell that these must be capable of existence because of the pigeonhole principle as it applies to compression, i.e. there exist possible systems which cannot be compressed to a more compact simulation. If they cannot be compressed to a more compact simulation, they cannot be reproduced except by exactly replicating every aspect of the system. If you exactly replicate every aspect of the system, you have by definition recreated the system, and are no longer predicting but merely observing the system. Entirely mechanically determined, noumenally unfree, but phenomenally free! Weird.

This is not even touching causality which, frankly, I lost track of once it got past the phrase 'necessary connexion'.

In this way I think we can relate the debate to something more than a disagreement over terms. If random events spark further events that affect your true essence, then you are not wholly causally determined, and some small degree of libertarian freedom is possible. Also, if the universe is wholly deterministic, then the nature of personhood and the nature of the Universe and the nature of the relationship of the one to the other affects whether we should accept hard determinism or compatibilism.