Yes. Once you start looking at this as a space, not just three orphaned ratios, a lot more structure pops out. The confusion matrix is basically a tiny algebraic terrarium for human error.
The most useful vocabulary clusters into a few families.
Everything starts from:
From there, most metrics are just different ways of slicing the same 2x2 box.
These ask: given what the world actually is, how did the classifier behave?
These are the “how good are you on actual positives vs actual negatives?” family.