An odd quantifier shift fallacy

I was flipping through the course packet for my Intro class this semester, and the professor included a chapter from some contemporary textbook. The chapter includes this paraphrase of Anselm's second ontological argument:

Premise 1: A being that cannot be thought of as not existing is greater than a being that can be thought of as not existing.

Therefore, if God can be thought of as not existing, then a greater being that cannot be thought of as not existing can be thought of.

Premise 2: God is the being than which nothing greater can be thought of.

Conclusion: God cannot be thought of as not existing.

Which is a little weird. But it's the ontological argument, so I guess weird shouldn't be a surprise.

Let Gxy abbreviate `x is greater than y' and Tx abbreviate `x can be thought of as not existing', and g name God.

(1) (x)(y)([-Tx & Ty] -> Gxy)
Therefore, Tg -> (Ex)(-Tx & Gxg)
(2) (x)(-Gxg)
Conclusion: -Tg

Even if we include both premisses in the inference to the `therefore', it's invalid. The hypothesis that God can be thought of as not existing does not imply that there is something which cannot be thought of as not existing. That is, a model in which only g exists, and g does T, is entirely consistent with (1).

One might try to read Anselm as instead arguing from

(1') (x)(Tx -> (Ey)(-Ty & Gyx))

Then the `therefore' follows by just instantiating the universal quantifier. Notice that (1') follows from the conjunction of (1) and

(1'') (x)(Tx -> (Ey)-Ty)

That is, `if something can be thought of as not existing, then something else cannot be thought of as not existing'. But what does the first clause have to do with the second? Maybe (just maybe) you can argue from the contingent-yet-actual existence of my chair to the existence of a necessary, first cause sort of being -- something had to cause my chair to be, and something had to cause that thing to be, and so on. But that won't work if the something is, say, my pet unicorn. Logically speaking, there's no reason to rule out the model with the empty domain -- in possible world terms, there's no reason given at this stage of the argument to rule out the possibility that nothing (not even God) exists.