The terms \'operator precedence\' and \'order of evaluation\' are very commonly used terms in programming and extremely important for a programmer to know. And, as far as I
The only way precedence influences order of evaluation is that it creates dependencies; otherwise the two are orthogonal. You've carefully chosen trivial examples where the dependencies created by precedence do end up fully defining order of evaluation, but this isn't generally true. And don't forget, either, that many expressions have two effects: they result in a value, and they have side effects. These two are no required to occur together, so even when dependencies force a specific order of evaluation, this is only the order of evaluation of the values; it has no effect on side effects.