Question

Hello,
I have a question regarding the ontological argument.
First I’ll describe the argument in its initial form, according to my understanding, briefly.
Premises: 1. Even an atheist can conceive of the greatest possible being.
2. Part of the measure of an object’s greatness is its existence (necessarily) or its non-existence.
Conclusion: the greatest possible being must exist (and indeed must exist necessarily).
There seems to be a possible refutation of the ontological argument. The moment an atheist conceives of the greatest possible being, since he starts from the basic premise that God does not exist, he is excluding the criterion of existence from the criterion of the greatness of the greatest possible being.
I am mainly trying to challenge the application of premise no. 2 in the case of premise no. 1. Is there a logical flaw in the refutation I proposed?