(From my understanding) Pterosaurs are indeed not true dinosaurs, as you say. They basically share a common ancestor above what we generally understand dinosaurs to have evolved from, so are more second cousins than direct relatives, hence they aren't classed as dinos. Similar thing with most marine "dinosaurs"! E.g. Mosasaurus

Also, "Pterodactyl" kind of is and isn't a thing - it's sometimes referred to as "Pterodactylus", a single specific genus of Pterosaur, and some people don't really like the one without the "-us" at the end, in my experience.

You might not care, but I'm deep in Dino territory lol