A recovery phrase is a set of words, usually twelve or twenty four, that a wallet creates at setup and from which any compatible wallet can rebuild the same keys. It restores access, not the device and not the coins, which never left the network. An extra passphrase, if one was set, has to be known as well. With a lost recovery phrase and a wallet that still works, the sensible move is to create a new wallet and send the coins across. With both gone, nothing can bring them back.
Hot wallet vs cold wallet is less a contest than a division of labour. Many players keep a hot wallet holding the amount meant for play and a cold one for anything held longer, moving coins between them when needed. An exchange account is neither, because the exchange holds the keys whatever its app looks like. Whichever pair is used, each wallet needs its own backup, and a small test transfer is worth sending first whenever coins move from cold to hot toward an address not used before.
A hot wallet is any crypto wallet whose keys live on a device connected to the internet, such as a phone app, a browser extension or a desktop program. It is quick to use, which suits deposits and withdrawals at a casino, and it is the kind most people start with. The connection is also its weakness. Malware, a fake extension or a request approved without reading can reach keys on a connected device, so a hot wallet is best kept for sums in active use rather than for savings.