You heard it right. Exchanges do not give you control of your Bitcoin. They hold the private keys, and you are storing your Bitcoin in someone else wallet. They send Bitcoin once you make a request. They have the option not to send your Bitcoin if they want. They ask for your details to store your own Bitcoin.

Electrum is a good wallet. But If you want to hold for a long time and plan to invest regularly, I suggest buying a hardware wallet. Another important thing is security. Do not fall for scams and phishing. Never share your seed phrases.