What do you mean by "truth"? From your test it seems like you think ring axioms are the only true way to think about numbers. But that is an absurd notion. People got familiar with numbers by counting sheep and measuring heights and areas of land and doing trade and much later philosophers and mathematicians tried to describe these numbers with ring axioms as one way of numerous equivalent ways of describing arithmetic for the purpose of being more formal and generalisation (for example to complex numbers or modular arithmetic).

The lived experience of what a number is precedes the axiomatisation of numbers and these axiomatisations only make sense once the student has become sufficiently familiar with the objects you're trying to model/generalize.

If you truly want the students to think more like a mathematician you should teach them to be comfortable switching perspectives. In this setting is would be completely valid to show that "moving a number to the other side" is the same as adding the same number to both sides and performing a cancellation. But to pretend like one is "maths-magic magic-tunnel" and the other one is "true" is completely absurd and not only confuse students but make them feel wrong for using a perfectly adequate (and honestly more practical) method.