I don't think coordinate systems are required to be independent or orthogonal.

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

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

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

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

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

Note that any combination of two of the three "dimensions" does constitute an independent system. That is, given that q + r + s = 0, then q + r = -s, so you could use only (q, r) as your coordinates (and indeed that's likely how you'd canonically refer to them)

The "third coordinate" is only for convenience when working with the three main axes of an hexagonal grid. Without it, the three directions of your grid are {+q, +r, -q-r}, which introduces an ugly asymmetry for one of the three axes.