Actually, I think all the examples you provided were independent dimensions. I'll concede orthogonality.

More specifically: can you think of coordinate systems with explicit binary or ternary constraints on the parameters in the coordinates? I can't immediately think of any.

Holonomic constraints etc. are constraints on the validity of regions of the space, not constraints on the ability of coordinates to exist.

Good point! Hmm...

If I understand correctly, homogeneous coordinates have additional independent dimensions, but are many-to-one instead of constraining their parameters. Maybe normalized homogeneous coordinates? That's not a particularly interesting system.

I'm now second guessing my definition of coordinate system...

Would points in a simplex qualify as a coordinate system? Their dimensions would have to sum to 1. That is, so-called "normalized" barycentric coordinates maybe, as opposed to standard barycentric coordinates?

https://en.wikipedia.org/wiki/Simplex

https://en.wikipedia.org/wiki/Barycentric_coordinate_system