Custodial means that a company owns your coins. You don't actually one the Bitcoin in the wallet.

Non custodial means that you own the Bitcoin and you own the private keys.

For alby, it's custodial but they plan on removing that so that you have to run your own lightning node to use it. It should work for now though.

I've never used green but if they give you a lightning address it should work for zaps.