You are right, but also it helps against double spending and makes all transactions authentic and secure.
You don't need to run a full node to verify whether a transaction is 1) valid or 2) included into a block.
The SPV protocol handles by storing only the block header and not the full blocks.
I will appreciate further explaination about this, which wallets are non-node wallets? Are non-node wallets the custodial wallets?
"non-node" walltes are wallets which do not keep a copy of the blockchain.
They are called lightweight wallets or SPV wallets.
Custodial wallets on the other hand, are wallets where not you, but a 3rd party has access to the private key (e.g. exchanges).
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.
They are definitely clients, yes.
And both of them can create private keys (and seeds) securely, indeed.