Hi, I'm sorry to bother, but I'm not able to locate the catalogue in creative menu or even by rolling a fisher. Could I get the summon commands or the summon names for each axolotl?

More Axolotl Variants API doesn't add a catalogue. For vanilla variants you just should use /summon minecraft: axolotl ~ ~ ~ {Variant:"minecraft:lucy"}

I thought it did, according to a youtube tutorial it does. is there a list of the commands to summon the modded variants?

More Axolotl mod adds the catalogue.
There's a wiki which explains how to use this mod. https://github.com/AkashiiKun/MoreAxolotlVariantsAPI-Common/wiki#summoning-command