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)
The two examples of an Hodler you’ve given us actually correct and that is how an hodler and non-hodler be viewed as far as the bitcoin hodling is concerned and who an hodler is suppose to be or suppose to possess to quantify themselves as one. Leaving your coin on a custodial exchange means you don’t have control of the money but anytime the you can access your money in those exchanges without any cause for alarm, you may still be regarded as one. A non-custodial wallet is best for holding your bitcoin but most people today don’t even fancy it and stick to their exchange due to high transaction fees of moving the coin to an exchange, even though an hodler is not suppose to do such when they’re not ready to sell their holdings off.