Univocity of 'exists(s)'?

Bill is much exercised by an argument of Peter van Inwagen that he summarises thus:

a. Number-words are univocal.
b. 'Exist(s)' is a number-word.
Therefore
c. 'Exist(s)' is univocal.

He explains that

(b) captures the Fregean claim that ". . . existence is analogous to number. Affirmation of existence is in fact nothing but denial of the number nought." (Foundations of Arithmetic, p. 65)

He goes on,

Consider my cat Max Black. I exclaim, 'Max exists!' My exclamation expresses a truth. Contrast the singular 'Max exists' with the general 'Cats exist.' I agree with van Inwagen that the general 'Cats exist' is equivalent to 'The number of cats is one or more.' But it is perfectly plain that the singular 'Max exists' is not equivalent to 'The number of Max is one or more.' For the right-hand-side of the equivalence is nonsense, hence necessarily neither true nor false.

So Bill thinks that 'exist(s)' in 'Max exists' has a different meaning to 'exist' in 'Cats exist', and he denies PVI's univocity claim.  After some more argumentation he concludes,

So if you want to maintain the univocity of 'exist(s)' across general and singular existentials, you must either conflate the 'is' of identity' with the 'is' of predication, or embrace haecceity properties.

Bill has always maintained that there is an important distinction to be drawn between what he calls 'singular existence', exemplified by such as 'Max exists', and 'general existence', as in 'Cats exist'.   For all the ten years or so I have been reading him I have yet to understand what he is getting at.  For suppose that Bill's cat Max is the only cat there is. Then Max is our 'witness' that there is at least one cat. That Max exists shows that cats exist.  And if cats exist then there is at least one cat, in this case exactly one, and Bill has chosen to call him 'Max'.  I can't see how the verb 'to exist' is being used differently in the two cases.

Unfortunately, PVI's argument weighing in on the side of this univocity thesis gives Bill a hostage.  For it allows him to come up with 'the number of Max is one or more' as a paraphrase of 'Max exists'.  Besides being ungrammatical English, Bill can argue that any attempt to make sense of this leads to unacceptable conclusions.  So PVI has scored an own-goal.

I argue for the univocity thesis as follows.  Observe what Bill says,

Consider my cat Max Black. I exclaim, 'Max exists!'

He says that his exclamation expresses a truth.  Indeed it does.  We can see it's true because his first sentence, which is actually imperative, tells us that Bill has a cat whom he calls 'Max'.  In other words, some cat, in fact the one Bill calls 'Max', exists.  Bill's exclamation merely confirms, or emphasises, what we know already, namely, that there is a cat and that Bill calls him 'Max'.  But this is a general existential claim. The singular claim 'Max exists' is really a disguised general claim.  I say that all existential claims are in fact general and the question of univocity does not arise.