I spend a lot of time looking at CAD models and assemblies with multiple coordinate systems, i.e. there's a global and then many of the parts have their own local, even with many instances of duplicate parts all rotated and shifted, each with their own local system, and I've never really thought about this as an issue!
I think in general when each part has its own coordinates one refers to partname->x, partname->y, partname->z (in much the same way you theaters use stage-left, stage-right and house-left, house-right.
For referring to a plane in a particular system it generally is partname->xy, partname->yz, and partname->xz where you'd refer to opposite faces either as + and - or by a coordinate reference on the third axis (e.g. partname->xz, y equals zero plane, partname->xz, y equals 1 plane). For something that's not a part (e.g. a graph of data) you'd use graphname instead of partname.
Things like top, bottom, front, back get really complicated because it's not unusual for the natural way to set a part on its own in gravity can be very different from how it's installed (with or without gravity) and what environment it might normally operate in (may or may not have gravity)
EDIT: to add the "y equals" to the description of plane naming