I agree, wyverns would be a cooler name. Just to clarify, in most legends/myths/game/other media:
1. Dragons - the name of the species
2. Wyrms - either ancient dragons in the older myths, also generally applies to non-oriental snake-like dragons in more modern media, which may or may not have wings or legs
3. Western Dragons - generally the type of dragon you see in movies, with four limbs and two wings, often stands on four feet regularly, but may also occasionally stand on two as its regular stance
4. Eastern Dragons - associated with rain and storms, not fire, these are snakelike dragons with manes and horns, and fly using telepathy
5. Wyverns - these are medium-sized (around adult human size) and walk on two legs. They are built like a bird, with two legs and two wings instead of arms, they sometimes can breath fir, but not always
6. Faire Dragons - These are tiny creatures that range in size from the size of a large butterfly to the size of a small dog, the amount of limbs can vary, but in most things these creatures have no limbs except for their large butterfly like wings, imagine a butterfly and replace the main body with a tiny snake body
Those are the five main types, though their are many others that are often more culture specific. I won't bore you with the knowledge of those though, so if you want to find out more, I suggest you look it up yourself.