A better privacy. You won't have to worry about third parties collecting information about you.
You are right, but also it helps against double spending and makes all transactions authentic and secure.
In addition to better privacy as mentioned above, running your own node also means you don't have to trust a third party. You can validate all blocks yourself, and you can verify every transaction you send or receive yourself. Any non-node wallet must connect to someone else's node, and trust that they are honestly verifying transactions and sending you the correct information.
I will appreciate further explaination about this, which wallets are non-node wallets? Are non-node wallets the custodial wallets?
Also, if I have an open source wallet like bitcoin core and eletrum, are they example of bitcoin client? As they are made of softwares that can generate private keys securely on their own.