Both counts, Wagen is just a even broader definition that counts for more kinds of vehicles.
I think it just got carried over from the time when carriages were still the main way of transportation and cars at the beginning were just motorized carriages. A lot of other, more specific terms from the time of carriages are still being used today, like „break“ (also for station wagon), Landaulet, Phaeton, Coupé etc..
But maybe it’s just a thing of language and what we are used to, it sounds just normal to us and fits the object we want to describe with it.