Stimulating as always! I have a criticism to make of the use made of the term 'rigid designation'.

Multiple philosophers have suggested that this stance seems similar to "rigid designation", i.e., when I say 'fair' it intrinsically, rigidly refers to something-to-do-with-equal-division. I confess I don't see it that way myself [...]

What philosophers of language ordinarily mean by calling a term a rigid designator is not that, considered purely syntactically, it intrinsically refers to anything. The property of being a rigid designator is something which can be possessed by an expression in use in a particular language-system. The distinction is between expressions-in-use whose reference we let vary across counterfactual scenarios (or 'possible worlds'), e.g. 'The first person to climb Everest', and those whose reference remains stable, e.g. 'George Washington', 'The sum of two and two'.

There is some controversy over how to apply the rigid/non-rigid distinction to general terms like 'fair' (or predicates like 'is fair') - cf. Scott Soames' book Beyond Rigidity - but I think the natural thing to say is that 'is fair' is rigid, since it is used to attribute the same property across counterfactual scenarios, in contrast with a predicate like 'possesses my favourite property'.