Now I'm properly confused. Are ions not atoms? Like, we call protons protons when we're talking nuclear and call it hydrogen when we're doing chemistry. What makes helium and alpha particles different? (Genuine question.)

Strictly speaking, atoms are neutral and ions are charged. Ions can be monoatomic (with a single nucleus, like He2+) or polyatomic (like H3O+).

H+ is tricky; it’s called hydron (or just H-plus) but it is indeed formed of a single proton, so in practice it is also called that. The names are interchangeable and depend on the field. For example, when looking at irradiation, we call it proton, but either is correct.

If we want to be pedantic, an alpha particle is a He2+ formed during a nuclear reaction, so it typically has quite a bit of kinetic energy as well. In that context they are interchangeable. I am not aware of He2+ being formed in other ways, He is very stable and I am not aware of something that is able to steal its electrons.