Anselm's attempted proof of the existence of God has often been criticised for the assumption that 'existence is a predicate'. Is that right? I'm going to have a stab here at representing the argument without using the word 'exists'.
The first thing to note is that Anselm gives two arguments, not one. The first is in Proslogion chapter 2, the second is shortly after in chapter 3. Another thing to note is that the argument was written in Latin, of a fiendishly difficult syntax, but I think the 1903 translation by Sidney Norton Deane is actually quite good, though not literal *.
Here is the argument in numbered steps, with minor changes that don't materially impact the sense of the original translation.
1. It is possible to conceive of a being X which cannot be conceived not to exist.
2. Such a being is greater than one which can be conceived not to exist.
3. Suppose that a being Y than which nothing greater can be conceived, can be conceived not to exist.
4. Then Y is not that, than which nothing greater can be conceived.
5. But this is an irreconcilable contradiction.
6. Therefore there is a being than which nothing greater can be conceived to exist, that it cannot even be conceived not to exist.
7. This being is you, O Lord, our God.
The real difficulty is the way that Anselm nests the verb 'conceive'. We are not just conceiving something, we are conceiving something as conceived. I will try and unpack this by representing the argument as a set of 5 steps, all of which fall within the scope of an overall 'conceive' that forms the main verb, and by translating 'x does not exist' as 'nothing is x' (thus removing 'ontology' from 'Anselm' - see the title of this post).
(0) We can conceive of an x and a y such that
(1) we cannot conceive that nothing is x
(2) nothing greater than y can be conceived
(3) we can conceive that nothing is y
(4) x is greater than y
(5) something greater than y can be conceived
Some comments. Step (3) is 'we can conceive that nothing is y' whereas strictly it should be 'nothing is y'. Anselm has "can be conceived not to exist" where the verb 'conceive' is clearly the one belonging in step (0). but then the argument is not valid. My translation is valid, however, if it follows from my conceiving that nothing is y, that by immediate reflection I can conceive of myself conceiving this, and thus conceive that I am conceive that nothing is y. Otherwise, this is a serious flaw in the argument.
For a similar reason, I move from step (4), "[We can conceive that] x is greater than y", to (5) "[We can conceive that] we can conceive that x is greater than y". If these steps are valid, it now follows that we are conceiving a contradiction, i.e. steps (2) and (5) contradict one another. Note, this does not involve an actual contradiction, only the conception or the thought of a contradiction. But this gets Anselm to where he wants, namely to a concept such that it is contradictory to think of it not being instantiated. The verb 'exist' does not appear anywhere, thus translated. The only question is the validity of the move from conceiving of an F, to conceiving of conceiving an F. Is it valid? I don't know.
* For example, he translates Nam potest cogitari esse aliquid, quod non possit cogitari non esse, which is literally, but clumsily "for it is possible that something is thought to be, which could not be thought not to be", as "For, it is possible to conceive of a being which cannot be conceived not to exist". I.e. he converts the passive to an active, and translates 'esse' as 'exist'.