I believe it's wyverns that have two legs, and are drakes not lesser dragons? Would not make sense to only have 2 legs in that case.
You're right. Wyverns have two legs. Dragons have four legs. Drake is literally another word for dragon; there is no distinction. I think in Dark Souls they associated drake and wyvern together.
In the japanese version they are referred to as wyverns, i believe the localization just changed them to the name of Drake, i guess to make it more understandable? but in actuality a Drake has 4 legs and no wings, Wyverns-2 legs + wings, dragons-4 legs+wings