The question you are asking is a social question, and it cannot have any answer better than a social answer.

Is there a terminology that the orbifold community has settled on and uses universally? No.

Are there common terminologies that are used by various members of the orbifold community? Yes, some of them are listed in other answers.

Given these social realities, how should one refer to a point of an orbifold with non-trivial stabilizer? Pick the one which makes most sense to you for your context. Or, as Misha suggests, pick the one that most honors the historical origins of the concept.