Translation - Japanese (literal translation)
Literal translations are for CLARIFICATION and not fucking suggestions for alternate translations like retards fucking assume every fucking time I do this.
Dragons - 竜 (ryuu = Dragon)
Ancient dragons - 古竜 or 古の竜 (koryuu/inishie no ryuu = Ancient Dragon)
Wyvern - 雷の飛竜 (ikazuchi no hiryuu =Flying Dragon of Lightning)
Hellkite Dragon - 飛竜ヘルカイト (hiryuu Herukaito = Flying Dragon: Hellkite)
"Drakes" mentioned in drake sword name/description, as well as the valley of drakes - 飛竜 (hiryuu = Flying Dragon)
Guardian Dragon - 護り竜 (mamori ryuu = Guardian Dragon)
Dragon aerie - 護り竜の巣 (mamori ryuu no su = Guardian Dragons' Nest)
"Wyrms" the dragonriders are said to have used - 地竜 (ｊiryuu = Earth Dragon)
Drakewing Ultra Greatsword - 翼竜の特大剣 (yokuryuu no tokudaiken = winged dragon's greatsword)
"Drake" as used in the drakewing ultra greatsword's description is also 飛竜.
Paledrake - 白竜 (hakuryuu = white dragon); Also "Seath the scaleless" is originally 白竜シース/White Dragon Seath.
Black Dragon - 黒竜 (kokuryuu = black dragon)
Slumbering Dragon - 眠り竜 (nemuriryuu = sleeping dragon)
Drakeblood - 竜血 (ryuuketsu = dragon blood)
Archdrake sect - 古竜院 (koryuuin = Ancient dragon('s) institution)
Archdrake - 古竜 (koryuu = Ancient Dragon)
Stone dragon/everlasting dragon - 石の古竜 (ishi no koryuu = Ancient dragon of stone)
Everlasting dragon (from the opening movie) - 朽ちぬ古竜 (kuchinu koryuu = Ancient dragons that would not die/decay/end; This is not a name for a type of dragons, it's a descriptor + "ancient dragons")
So yeah, the "wyverns" and "hellkite dragon" and "drakes" are all in fact the same sub-species of dragon, the hiryuu ("flying dragon"), which is a common Japanese translation (alongside 翼竜/yokuryuu/winged dragon) for English wyvern or drake. They're all offshoots: The "wyverns" are in fact a type of hiryuu that has something to do with lightning (hence them shooting lightning at you), while the Hellkite is either the name of a specific specimen, or another different offshoot of the sub-species.
"Guardian dragon" seems to be more of a description of their
role (guarding the ancient dragon's home) than of their species name, and the sword seems to indicate that they're also hiryuu. Similar, in the cases of the black, white, and sleeping dragons, these are clearly titles, rather than the types of dragons they are.
The dragonriders' jiryuu (earth dragons) are seemingly another, completely different type of dragon (there aren't many details on the things).
So yeah, in Japanese they're all dragons; The Ancient Dragons are differentiated from other dragons by always being referred to as such, except in the cases where they have a title prioritized over it.
Hiryuu is almost always translated to "drake" but it looks like the confusion comes from how the translation is somewhat inconsistent (hellkite DRAGON, and the ikazuchi no hiryuu ought to have been Thunderdrakes or something instead of an entirely different word). And I can sort of see what they were trying to do with jiryuu to wyrm: Wyrm sounds like worm and worms are associated with Earth, right? But people familiar with fantasy take "wyrm" to just mean "dragon" which had people thinking the dragon riders rode drakes of maybe even actual dragons when they in fact rode big lizards that crawled around (see the concept art) which is something that comes across fine in the original name (they're EARTH dragons, as opposed to FLYING ones).
There also seems to be some confusion stemming from the whole ancient dragon/everlasting dragon/archdrake thing. All of these are just "ancient dragon" in Japanese, with "everlasting" being nothing more than a descriptor for the ancient dragons.
tl;dr The translation job here is all just very inconsistent and confusing.