Transaction cancelled due to being out of gas. I'm freaking out!

Hello people

About an hours ago, I've made a transaction trying to move all my ETH from one exchange to another. Pretty quickly, I saw on the target exchange that the transaction was cancelled due to being out of gas.

So I've read a bit about this and apparently when this happens, the ETH (minus the gas burnt) is supposed to move back to the original address, but on the original exchange, my is still showing as 0!

Did I lose this ETH? What can be done in a situation like this?

UPDATE: As u/uwu2420 pointed out, apparently the target address is a smart contract, which is why the transaction failed (due to requiring a slightly higher gas amount). The exchange got back to me saying they would return my ETH.

Thank you all for the help, really appreciate it!