I used to always refer to the Envoy as the Black-Cloaked Envoy or the BCE, but now just "the Envoy" seems sufficient a lot of the time -- which is definitely a familiarity thing.

Same here. I still use "BCE" or even the full "Black-Cloaked Envoy" when I want to emphasize the (political or quasi-mythical) weight of the role, but otherwise ...