Different error messages under "Mandatory script verify flag failed."
But aren't both errors just variant of a failure in executing the OP_EQUALVERIFY? At what step of the evaluation each error is thrown and what can we make out of it?


No, they're different errors. There are two rules that must be satisifed here:


The hash of the public key must match a certain value. (A failure here means you have the wrong key, or no key.)
The signature must be valid for the message and public key. (OP_CHECKSIG does not error immediately. It returns 0, which causes the script to fail.)


There are other script errors, too. See this list: https://github.com/bitcoin/bitcoin/blob/master/src/script/script_error.cpp