Not the official definition of balance, just my opinion.
I think it is not good to measure a card balance by use rate without comparison to the card’s alternatives.
E.g. if Zap is the only small spell in the game, even with a usage way over 50%, it doesn’t mean the spell is OP. It is only overused because there is no alternative.
When looking at usage rates, I think it only make sense when reviewing in a group of similar cards. E.g. win condition, range, splash, mini tank, tank killer, cycle, swarm, small spell, big spell, spawner, siege.
Then perform a t-test or chi-square test with the usage stats within the group to see if there is any card that has significantly different usage rate.
It can also apply to win rates. But both overall win rate and comparison with similar cards are important.
I think the statistics part is basic. The difficult part is how to balance and how to handle players’ expectations.
Rum Ham's general policy for balanced cards is, as you said, 9% UR, 45-55% WR. However, there are certain "niche" cards that have lower use rates but higher win rates. Those imply cards that require certain synergies to be viable (like night witch or sparky) or have a higher skill floor or both. Those would be considered balanced.
I personally don't think use rates and win rates should be the end all be all. If a card is slightly above or slightly below the standards it's not terribly unbalanced. When it gets too high or too low is when balance needs to be done (adjust the most OP and most UP, leave the slightly OP or UP cards alone because even small changes could break or kill the card).
Also remember that RoyaleAPI does not tell the full story, only Supercell has the "true" stats. Generally, no card has a 60% or more win rate or less than 30% win rate. I'd say 14-days is a better indicator than 1-day though