That's spot on, mate! SPV wallets do indeed rely on the full nodes of others to verify transactions. They're a lighter-weight solution for folks who can't or don't want to run a full node.

As for Lightning Network wallets, they're considered "hot" because they are connected to the internet and used for active trading and transactions. Being off-chain doesn't really affect their "temperature". The main thing is whether or not they're online all the time – that's what makes them hot!