Types of wallet
Desktop
You download the blockchain and have a copy of the keys and wallet address.
Everything checks except here. There are 2 types of Desktop wallets. One is as you described "full client" that downloads whole blockchain but there are some wallets that are SPV wallets. They don't need to keep whole blockchain, only fraction of it to be able serve as wallet Desktop
You download the blockchain and have a copy of the keys and wallet address.
. SPV clients are multibit and electrum.