Now here comes my question: If those who buy and hodl their coins on a noncustodial wallet where they have full control over the coin are referred to as hodlers, are those who also hodl under a custodial service where they don’t have full control of the coins also regarded as hodlers?
If you leave your coins on a custodial exchange, the exchange is the one holding your coins and not you. What's available in your balance is what they can manually input. Since you don't have full control of the coin, you shouldn't be considered a hodler.
Not your keys, not your coins (and not being a HODLer)
Your keys, your coins (and you are HODLer)
You came up with a very good analogy to the later.
Now it’s important to note that, hodling actually comes with a responsibility of having these coins kept an away from any form of trading or providing liquidity.
If you aren’t having authorization on the utility of your coin, you are not a hodler.
Should you have your coins on an exchange, you aren’t holding Bitcoin simply because, you can’t store your coins on an exchange and the exchange could always do away with you coins. So no, the term hodler doesn’t apply to those who have their coins on some exchange.