So balance =/= balance, and that's the accepted definition. If there's a range, then it's not balanced. Period. Try balancing a pencil on your finger. If it's not equal on both sides, it will fall because it's NOT BALANCED. Balance = equal.
No, it doesn't, and it
literally never has. When you "balance" a chemical equation, you don't make it so that both sides are literally identical things--you make it so the charges and masses are equivalent, but redistributed. When you "balance" a checkbook, you don't ensure that the money paid is precisely equal to the money present, you simply make sure that every payment is covered. When you "balance" a set of scales, you don't put
literally identical objects on both dishes, you put an object to be examined and the pre-weighted objects you measure with until you get something immesurably different from true equilibrium, but
the whole point is that you're comparing two things that are different, not identical. Even in algebra, where you
are trying to generate exact equalities, the things involved need to (at the very least) be different in
form for the result to be relevant. "Balancing" where you end up with "2=2" is trivial, and tells you nothing (with the possible exception that you started from two equations that were not linearly independent...which still tells you nothing you couldn't have known before).
You can insist that "balance = equal" all you like. Dictionaries, textbooks, and the general public all disagree with you. And with that, I'm absolutely, positively done discussing the topic, because there is
literally no point in having a debate with you if you're going to dogmatically insist that a word means a thing that
it simply does not mean.