When you keep your coins on exchange, you don't have access to them, you just see the numbers, that's all. You can't sign a message, you can't export keys, you can't do anything, so you don't HOLD or HODL coins in this case.
You only HODL coins when you have them in your hardware or software wallet, for example you are a HODLer if you own coins on your Electrum wallet, if you own coins on your ColdCard wallet and so on.

Not your keys, not your coins (and not being a HODLer)
Your keys, your coins (and you are HODLer)

Not your keys, not your coins. If you don’t have the full control over your private keys, then it’s actually not your coins. In short, you don’t own your coins but the exchange holds the full ownership. They are the one hodling your coins, not really you.

What you only have is just a piece of paper containing your coins your owned, but in reality the custodial exchange controls everything. If the exchange goes bankrupt and suddenly stop its operation, you could end up losing your access and thus, losing your funds in the exchange.

You are right, and I agree with you! Someone who keeps their crypto coins on an exchange shouldnt really be called a "hodler," at least not in a serious way. But, you know, it kind of depends on how we define "hodler."

If we think of a "hodler" as someone who saves, but specifically with Bitcoin, then we can compare it to how saving works in regular money. You can save money by keeping it at home, like in a piggy bank or a safe, and you can also save it by putting it into a savings account at a bank or another place that holds money for you.

From a quick Google search the oldest and still running exchange is bitstamp and I'm pretty sure some of the OG's of this forum has little bitcoin in this exchange till date from the 2011's and perhaps if truly someone has any amount of bitcoin in his bitstamp exchange account or wallet till date and still holds it in this exchange would you consider the person a Hodler?

From your definition of Hodler they are not but if those assumptions are true then I would consider someone who has bitcoin from them till date a hodler even though we may not agree to it.

Just an assumption it may be true or not that someone hodls bitcoin in his bitstamp wallet

A share of the exchange holdings.

You have full control of your coins on exchange as long as your contract with the exchange still exists and the exchange is in service, but then why we refer to coin on exchange out of the holder's control is because exchanges most especially the centralized exchanges are subject to regulations and most times accounts get locked due to regulatory compliance.
This characteristic is what makes holding your assets risky but aside from that your Bitcoin on the exchange has all the other features of Bitcoin originally, what makes one a holdler is the amount of duration of time that they can hold on to their Bitcoin that is what differentiates us from traders.

No you don’t. Having full control of one’s coins means you own the keys. Centralized exchanges do not give keys nor seed phrase to their users instead all you get is a password and username. Storing your coins on exchanges simply means that you’re trusting a third party with control of your coins.

The fact that you are the owner remains you as the HODLer, and not the exchange itself. But if we are talking here the safety and security of your coins, I doubt it’s not 100% since you share your private keys with the exchange, which could put your coins at a high risk of losing especially when sudden circumstances happen to the exchange.

In the past 15 years we've seen so many exchanges open and shut down that there's nothing more insecure than keeping coins on an exchange. Those who hold their coins on exchanges aren't really hodlers, maybe we can call them forgetful. Even the biggest exchanges have either pulled exit scams, been forced to shut down due to legal issues or lost users' funds to hacks. Never trust any exchange. Always keep your Bitcoin in your own wallet.

They should definitely be called holders because in answer to what is a holder, we know that those who hold various types of coins for a long time. We call holding the effort to hold them for a long time by buying them at a lower price and selling them at a higher price. And we know those who do this holding as holders. A person can deposit their coins in CEX and (where they deposit their coins is a personal matter). However, no matter where they hold them, the coins will be held and those who want to hold these coins are called holders.

Exchanges can be scammed or hacked and closed, but holding coins is a type of holding. There are many exchanges where many people have held coins for a long time. So what would you say to those who have held tokens on exchanges for a long time? I think there can be any medium to hold tokens. No matter how you hold the coin, it will be holding and those who hold it are called holders.

However, you can say here that exchanges are not safe for holding tokens/coins but holding coins on exchanges is definitely holding. We cannot trust it due to the bad condition of the exchanges but of course keeping our coins in wallets is most important for our security.