I think the correct proposition is
(∀X)[X’s essence is described as “the thing than which no greater can be thought” → X exists in reality].
Now we ask: to what does X refer? If it refers to an idea of TW, then an idea cannot exist in reality. If it refers to a real thing, then the proposition is a tautology and proves nothing.
Perhaps N (renamed from X) refers to the essence or form of the being than which no greater can be thought. N is a description, piece of information. Let me then rephrase it as a property P = “such that no greater can be thought of it,” where “it” is whatever object has this property. Object B has property P whenever (∀C: C is an idea)[C ≤ B’s essence].
Now there is the world of ideas and real world. In the former, there are thoughts. Let one such thought be X. When thinking, we can conceive of beings. One such is B. These beings can have properties, including P. So, the full “path” is World_of_Ideas.X.B.P. There also can be a Real_World.B.P.
The first B is an idea, conception. The second B is real God. They are not the same, just as file C:\World_of_Ideas\X\B.txt would be distinct from file C:\Real_World\B.txt.
Suppose there is no second B with P. There is no object in the real world than which no greater can be conceived. We can’t say: ah, we have found a World_of_Ideas.Y.C, where Y is another thought, that is greater than the World_of_Ideas.X.B. This C has P and moreover, there is Real_World.C.P. The two Cs are distinct despite having the same property.
What if we say that P is a peculiar property which says that in the entire “filesystem,” there can be only one C? Again this makes no sense, because an idea of God cannot be identical to God.
Again, let N refer to an essence. Define “greatest essence” counterfactually to be that description P which, if a real object had it, it would be the greatest possible thing (or TW).
Let World_of_Ideas.X.B answer to P. Then B is a conception of that thing that, if it existed in reality, then it would greatest. This at least has no hidden equivocations, but I don’t know how to generate from it an argument for the existence of God.