Some investors are actually very reluctant and fear for nothing, which is why they prefer leaving their coins in the exchange instead of holding them in a private wallet. They consider self-custody as something with too much risk, wallet protection, and they don't trust themselves much that they can safely keep their crypto themselves, and also the idea that some exchanges sell to them that even if they get compromised, their coins are safe with them and will be replaced, gives them the safe mind to trust third parties for long, even when they shouldn't have.

As much as I strongly advice everyone against such practice as keeping ones coins or bitcoin rather, on a centralized exchange, I still will say that a holdler is a holdler, regardless of where they choose to hold their bitcoin, there is no much difference from the investor who choose to keep his bitcoin on a self custody wallet and the one who choose to keep his one centralized exchange, the only difference is that one person has access to his or her wallet keys and as such, have full control over the bitcoin, while the other does not have access to the keys to the wallet where his bitcoins are held and as such, have limited access to the bitcoin.

But facts remains that which ever amongst this two that want to sell can sell, the one holding in a self custody wallet will have to first move the fund to an exchange where he can sell it, while the other person who already have his funds on the exchange will simply sell.

Aside the fact that centralized exchanges can be hacked, go bankrupt, turn into a rug/scam and other risks that are completely out of the control of the user, one saves money on fees when bitcoin to be sold is already on the exchange.

Who exactly should be referred to as a HODLER?

As we already know, hodlers are people who hodl bitcoin for long periods of time; they are determined not to sell for the long term irrespective of the price change.

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.

It's not an investor, because if they are investors know how to store assets or wealth into a personal wallet. It's nonsense to store assets on the exchange for a long time because of fear and lack of confidence. How many times have major exchanges experienced theft or been hacked? and still believe they hold your assets? seriously? come on the purpose of bitcoin is so that you become the sole controller of what you have and this is also in line with the main goal of an investor.

It's just that some think hodlers are only those that hold for long enough, some years or so, whereas others -  they are traders or not worth considering.