The label role means the same thing on a boundary=administrative relation as it does on a boundary=place relation. admin_centre is for clarifying that one particular place within an administrative area’s boundary serves as the administrative area’s administrative center (in other words, the seat of government or capital city).
A label necessarily lies within the boundary, except in some rare cases where the boundary has been modified to specifically exclude the commonly understood center. An admin_centre usually lies within the boundary, except in some cases where a neighboring boundary serves as the seat of government.
I agree that both label and admin_centre are confusingly named. I think label was originally coined as a hint to renderers, as an override for when the default centroid calculation looks wonky, but that isn’t generally what it’s used for today. Meanwhile, admin_centre gets misused a lot because mappers think it means “the center of the admin[istrative boundary]” rather than “the center of admin[istration]”. Remember this point the next time someone promotes a shorter tag spelling because it requires less typing.
There have been proposals to rename these roles, but so far nothing has caught on. The existing roles have already “found support” among data consumers. The root of the problem is that we’re talking about two different notions of a “center” that can exist simultaneously: the cultural/demographic/economic/logical center of the place itself, versus a different place that serves as the place’s administrative center. It’s difficult to describe these relationships succinctly.
Technically, admin_centre isn’t really a geospatial relationship. Rather, it falls in the same category of basic place facts as a country’s flag or coat of arms, or a city’s twin cities/sister cities. But there’s a natural tendency to want to map this kind of information. If we say a place is a capital, then naturally the next question is what the place is the capital of.