It's the other way around - your credit score is higher if you have a lower balance on your cards.

That is not correct. A small balance raises your score more than a zero balance. But you want to keep it under a third.

This is an oft passed around urban legend, but it's not true. You can try for yourself by opening up a free CreditKarma account and seeing what happens if you zero out all your balances.

I'd take that with an entire shaker of salt; you are assuming CreditKarma knows the FICO formulas. Even the prediction engines used by the actual credit bureaus are notoriously unreliable at telling you what will happen given a particular change in inputs. Probably not accidental, they don't want to expose the algorithm.