Do I need (should) keep the cryptos in other wallet like MTW or MyCrypto? Will my balance on exchange become zero if I move? I am not sure about the procedure of that.
If the coin is transferred from the Binance exchange to another wallet, then your balance will be empty.
You really need a personal wallet that you can control yourself and without the interference of other people.
I prefer to keep my coins in MEW, METAMASK, TRUSWALLET. And I have never used any other wallet.