Is it accurate to say it is "not allowed" to divide both sides by something that includes a variable unless I can prove some other way that the denominator I'm creating can not result in 0?

In other words, I could divide both sides by a finite number or pi or something, but probably not by the variable I'm trying to solve for?

Is there a name for this rule perhaps?

To answer this, it is important to remember what any of this even means. What do we mean by any equation, like x-2=3?

Actually, without any context, this doesn't mean anything at all. But usually when we write "x-2=3" this is like a short-hand for a question. Namely "What are all of the values of x which satisfy the equation x-2=3?" Clearly in these very simple contexts, there is only one answer, and therefore we do not need to be very careful.

But now consider the example equation x*(x-1)=0. If we say "what are all of the values of x which satisfy the equation?" then you will lose a solution if you divide by x. Because if you do that, you get x-1 = 0 which has only one solution.

So the reason why it is invalid to divide by a quantity that "might" be zero is that it causes you to lose solutions, when the question you are trying to answer is "What are ALL of the values of the variable which satisfy the equation?"

To my knowledge this rule does not have a name. Rather, to talk about it, we merely point out that division by zero could be a problem, so we proceed to separate the solution method into cases. Each case avoids the issue of dividing by zero.

Alternately you can characterize this by the zero-product property: https://en.wikipedia.org/wiki/Zero-product_property

Note that in the zero-product property, you do not know which of the two factors is zero. Therefore to include all possible solutions to the equation, you must consider the possibility that the first factor is zero, and the possibility that the second factor is zero. If you do not consider both then you may lose solutions.

"it is invalid to divide by a quantity that "might" be zero [because] it [can cause] you to lose solutions"

^^ +1 THIS ^^

I wonder if it's just zero, or if it's any unknown value....

Dividing by an indeterminate value [can | does always] collapse a dimension of the problem, and so presents fewer solutions.

Also I will admit that my reptile brain sees simple equations like that and thinks what "IS" x, not "what ARE ALL POSSIBILITIES for X". The latter is completely my shortcoming. Maybe replacing what IS TRUE with what ARE ALL POSSIBLE TRUTHS is something I should do by default in life.

As an experiment, I've asked a few random adults x*x=4, what is x ,and only one of them acknowledged the possibility of -2 without prodding. I wish more ticktocks exposed mathematical truths like this than people trying to get a towel out of a running sink without getting wet...

It's only zero. If ax=b and a is not zero, then x=b/a. This effectively guarantees that division by a nonzero number preserves any solution, regardless of what x is (or what a and b are, again assuming a is nonzero).

This comes down to the definition of division. What does 15/5 = 3 mean? It means THE UNIQUE number x which satisfies 3x=5. There is only one number corresponding to division by a nonzero number.

What about 15/0? Well, we would need a number satisfying 0x=15. There is no such number and this is why so-to-speak "you cannot divide by zero".