About the "standing on 4 legs or 2 legs" argument, doesn't Kalameet stand on 2 when using his telepathic attack? I doubt sitting or wings have much to do with classification as an ancient or everlasting dragon.
If you start with the idea that a dragon has 4 limbs plus wings, you already cross out drakes and wyverns. From there, you do see almost every Dark Souls dragon is unique. A common trait of Everlasting Dragons is their scales and stone-like features and their mix of both organic and inorganic characteristics. Ash Lake's Stone Dragon seems to be everlasting considering his complete lack of reaction when you cut his tail, although he clearly is alive in a way, as you can see when he opens his wings as you approach him the first time. Also note the difference between "everlasting", which refers to time, and "invulnerable", which they are not.
As for the origin of Seath, the way the lore goes in the original game's intro, I think it would fit better if Seath was born/created slightly after the appearance if the First Flame, as this is what created disparity. The lore is quite clear that before the flame, the Archtrees and dragons were not varied. I think Seath might have been born/created this way not because of weird breeding, but simply as one of the many changes the flame brought to the world of Ancients.