Where off-chain transactions are registered afterwards?
It doesn't.

From the point of view of the blockchain there are just 2 transactions: one going into an exchange address the other is going out of an exchange address.

For example:


You sent 0.1 btc to an exchange [recorded on blockchain]
Some other people send other amounts to that exchange (exchange controls all these addresses) [recorded on blockchain]
You make 0.05 btc on a trade (that means somebody else lost that 0.05 btc) [NOT recorded on blockchain]
You withdraw 0.15 btc minus any trading fees
Exchange sends yours 0.1 btc + somebody else's 0.05 btc to you (they might use a couple of addresses that they sent from, or they might use a cold storage address) [recorded on blockchain]


The only transactions that are recorded on the blockchain are 0.1 btc from you, other deposits from other people, and you withdrawal of 0.15 btc as per example above. Any trades that are happening on the exchange are happening in their own database.

Also exchanges might consolidate balances into hot or cold address. So you might notice that after you sent 0.1 btc to an address that an exchange provided to you, that 0.1 immediately left that address to an address that has a large balance, but your funds are still displaying on your exchanges's account. That is because it is now handled by their internal database.