Your correct the two are different but a little bit interchangeable, just simply treat exchanges as a market place where you buy stuffs and wallet as your home where your stuffs are securely stored in. Exchanges can also be used to actually store your bitcoin but it’s the worst place to do that.

For example exchange acts as a marketplace so if you buy goods from the market and leaves it with the shop keeper you are not going to be in total control of those goods as you’re at the mercy of the shop keeper to give you access to them. But if you carry those goods to your home after buying, in this context you move bitcoin from exchange to your wallet, the wallet keys are with you and you have full access to your bitcoin and can do anything you like with it just like the goods you have in your house instead of waiting for the middle man.

So for full custodianship it is best to have your bitcoin moved to wallet which provides you with keys that you can use to access it anytime.