So because the numbers were defined with eight examples, no example explicitly showing associativity or commutivity, it's hard to see why there's no license to arbitrarily choose a modulus for each number?

Or perhaps we only feel like we can do that if that would let us make two sides of an equation equal? As if the implicit rule connoted by the examples was "if two sides of an equation can be interpreted as "equal", one must declare them "equal", where "equal" is defined as amounting to the same, whatever modular operations must be done to make it so? So the definitions are incomplete without an example of something that does not equal something else?