Quote
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 
.
. If you are looking for really simple definitions of things like wallet, keys, address, and SPV I'd like to invite you to check out my passion project. It's called Decryptionary and it's the simplest crypto dictionary.
You can find a simple definition of SPV at this link:
https://decryptionary.com/dictionary/simplified-payment-verification/
If you're a beginner, I recommend you check out my walkthroughs. They'll introduce you to crypto, show you how to buy your first coins and keep them safe.
https://decryptionary.com/what-is-cryptocurrency/