It’s just down to how you describe an animal. In pictures you show but in books it’s just description so it’s good to have somekind of standard to follow to help the reader understand.
Like a wyvern has a much different way of walking since it doesn’t have fore-limbs, and has to walks on the ends of it’s wings. Imagine a bat walking on all fours. A dragon has it’s wings on it’s back and walks on its 4 legs. By having a kind of standardized naming provides a much different mental image of how the creature behaves.
I am more talking about modern fantasy and not historical fiction/exaggeration.
Language evolves and there is no reason that we can’t invent new definitions/classifications of things that don’t reflect previous nomenclature. Especially for fantasy where everything is made up.
I agree there is no one true definition but I think something like this is the accepted descriptions that is also reflected in other media like DnD. For example, alot of people say Skyrim got their dragons wrong (naming wise) because they are more akin to what people consider Wyverns.
https://dragons.fandom.com/wiki/Types_of_Dragons
Edit: Added (naming wise)