~ a desktop wallet is a hot wallet you use for spending and have some small amount of bitcoin ready and fast accessible.
~ an exchange is only a place to keep coins as long as you are trading them (if they are in open orders) not just sitting in your wallet.
~ a cold storage should be used for all your coins (even if you are for some reason holding altcoins!)
~ web wallets are not at all recommended except if you are new to bitcoin and want to have the same feeling of "accounts" like PayPal,... to learn things first then move on to a decent wallet.
+1 for nothing online/in exchange except for brief pass-through's during a transaction to another party. Once you learn to store your coins in a cold wallet or use a hardware wallet you can have as many wallets as you are willing to KNOW the restore seed for. I use multiple wallets but I take responsibility for being able to easily restore them if something goes wrong on the device currently used to display them.