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Board Thread:Series general discussion/@comment-2112031-20170322125406/@comment-2112031-20170407102909
Given how contact between the various Assassin branches and Templar Rites ranged from limited to non-existant, it wouldn't at all be surprising if different groups went by different names.
The Roman Assassins called themselves the Liberalis Circulum, but we don't really know about the rest. According to the Encyclopedia, Aquilus was a member of the "Assassin Brotherhood from Lugdunum", which isn't helpful as it also refers to the Levantine Brotherhood as the "Assassin Brotherhood in the Levant".