Are there Wallets in an exchange?

Insider knowledge wanted. This is not specific to Binance but to all exchanges:

Do they really buy/sell crypto for me or do they just write all the trades in a database and account it the moment I withdraw.

For an exchange it must be far more efficient to just hold a pile of ETH, BTC,.... and just write down all the trades between the customers. No need to do real transactions on a blockchain as long as the volume does not exceed their reserve.

They would earn a shitload of money because they could keep all the gasfees.

I know from Safemoon that trading on most exchanges does not trigger the smart contract.

But is it common practice for all the other coins/tokens?