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)
I think you have already answered your questions. I don't think using a custodial wallet to hold Bitcoin can be considered as hodling because this is very risky. As long as one is investing his or her Bitcoin on an exchange, it means the exchange is holding it, and the exchange has the main seed to the wallet. Hodling refers to using the right wallet, like a non-custodial wallet, to invest in Bitcoin. If it is anything different from a non-custodial wallet, then one is risking money. The exchange is not safe, and anything can happen to the exchange at any time, which could prevent you from accessing your Bitcoin. When investing with a non-custodial wallet, your Bitcoin will always be yours, and you will always reach the destination of your investment, unless you are careless with your seed phrase, which someone else could use to access your wallet.