How is being an agent capable of making choices without outside influence compatible with determinism? My understanding is nobody can ever be an agent capable of making choices without outside influence. A person is fundamentally influenced at birth by external forces and moulding the person by deterministic process without the person having any real control. Furthermore one could only say they had free will if before life one agreed to take role of birth as a specific person and with knowing how life will play out till the end. Compatibilists are people who don't understand determinism because free will doesn't work with determinism. /hard determinist.
For most people, "outside influence" means, "at this point in time, only the contents of my brain affect the choice I make" and "no force or agent outside myself is compelling me to make a different choice than the one I'd make otherwise".
It's a word game of course. Even if you try to be extremely precise with your definitions, to avoid this kind of thing, most people won't quite follow.
Compatibilists believe what you believe. They just agree to use the popular, vague definition of free will instead of the strict one you're using.