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.