Here’s the thing. You said a “wyvern is a dragon.”
Is it in the same family? Yes. No one’s arguing that.
As someone who is an arcanist who studies wyverns, I am telling you, specifically, in arcana, no one calls wyverns dragons. If you want to be “specific” like you said, then you shouldn’t either. They’re not the same thing.
If you’re saying “dragon family” you’re referring to the taxonomic grouping of Draconae, which includes things from guard drakes to dragon turtles to pseudodragons.
So your reasoning for calling a wyvern a dragon is because random people “call all winged serpents dragons?” Let’s get guard drakes and wyrmlings in there, then, too.
Also, calling someone a human or an ape? It’s not one or the other, that’s not how taxonomy works. They’re both. A wyvern is a wyvern and a member of the dragon family. But that’s not what you said. You said a wyvern is a dragon, which is not true unless you’re okay with calling all members of the dragon family dragons, which means you’d call pseudodragons, drakes, and other winged serpents dragons, too. Which you said you don’t.
It’s okay to just admit you’re wrong, you know?