Ah, that does clarify a lot. You’re wanting a term to refer to the Xbox and Playstation class devices since they’re in this closely competing realm, but people already have a preconceived notion of what a “console” is (I would consider handheld to be consoles, for example).
It does seem like it would be useful to have a term for those.
If it’s a subcategory of console then it is a console. Otherwise it’s like saying a square is not a rectangle because it is a square, or a cat is not a mammal because it is a cat.
That’s why I say you are searching for a term for things like the Xbox and Playstation - because it would be useful to have a term for that. Simply “console” is not a good fit imo, but you may disagree.