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.
I agree that storing bitcoin on centralized exchanges or on 3rd party platforms is very risky and a bad idea but I still think they are holders. If they are not the holders, who are they and what shall we call them?
The concept of “holder” is very simple and anyone who holds any asset like stocks, real estate, cryptocurrencies…are called holders. The concept has been around since the 17th century and started to be used commonly in the financial world in the 18th century, while bitcoin and bitcoin investors have only been around for less than 2 decades. Cryptocurrencies are just a small part of a financial market worth hundreds of trillions of dollars. What right do they have to change this concept?