Annulus region masks restricted to mode='center'
Currently annulus regions (e.g. CircleAnnulusPixelRegion) are implemented as Compound regions and Compound region masks are restricted to use only mode='center' (see https://github.com/astropy/regions/blob/main/regions/core/compound.py#L68). This makes sense for generic user-defined generic regions, but not for the provided CircleAnnulusPixelRegion, EllipseAnnulusPixelRegion, and RectangleAnnulusPixelRegion objects for which the other modes are well-defined. For example, it's not currently possible to perform "exact" aperture photometry with any of these provided annulus region objects.
I thought that originally the annulus region masks could use other overlap modes (e.g., photutils provides this), so I suspect that this behavior changed at some point when the annulus regions were refactored. Perhaps it was unintended.