I suspect if you are running off a different calendar than most people you interact with then you would be used to clarifying these things!

Exactly the point, requires clarification, doesn't stand on it's own, is hardly unambiguous.

Moreover most people are surrounded by by people using the same calendar and don't clarify, it's an issue for travellers and data reconciliation across boundaries.

That said, it's the month and year that are most lacking from a simple { Dayname, day of month } pairing.

> Sorry, of which month in which year under what calender?

Whichever could be a theoretical possibility for whatever you're describing the date of. In almost all cases, there will only be one choice. But in other cases, the speaker will provide the rest of the date.