Kant is on to something here. If existence is a property of things, it is a rather peculiar one: you can find a blue marble, and also a non-blue marble (a red one, say), but you cannot find a nonexistent marble—a marble that lacks the property of existing. Of course, that does not mean Kant is right: a peculiar property is still a property. And in fact, according to many philosophers, Kant is wrong: existence is indeed a property, albeit a very undiscriminating one, because everything has it.
Kant is on to something, but Byrne is nowhere near it (and Kant's argument is a much better one than the one Byrne prefers in the next few paragraphs, which builds on a misreading of Anselm, and to which even Descartes, more vulnerable to this sort of objection, has an answer). There is nothing especially peculiar about taking existence to be a property, and Kant didn't think that there was. Kant's argument is that it is not coherent to take existence as a predicate explicating a feature of a concept, in the way that "having three angles" is a predicate explicating the concept "triangle" in the judgment "A triangle has three angles." His argument for this is quite clever (CPR A599/B627, Kemp, tr.):
'Being' is obviously not a real predicate; that is, it is not a concept of something which could be added to the concept of a thing. It is merely the positing of a thing, or of certain determinations, as existing in themselves. Logically, it is merely the copula of a judgment. The proposition, 'God is omnipotent', contains two concepts, each of which has its object -- God and omnipotence. The small word 'is' adds no new predicate, but only serves to posit the predicate in its relation to the subject. If, now, we take the subject (God) with all its predicates (among which is omnipotence), and say 'God is', or 'There is a God', we attach no new predicate to the concept of God, but only posit the subject in itself with all its predicates, and indeed posit it as being an object that stands in relation to my concept. The content of both must be one and the same; nothing can have been added to the concept, which expresses merely what is possible, by my thinking its object (through the expression 'it is') as given absolutely. Otherwise stated, the real contains no more than the merely possible.
Suppose you and I are arguing about whether I have a hundred-dollar bill in my piggy bank: you say, "There is a hundred-dollar bill in your piggy bank," and I say, "There is no hundred-dollar bill in my piggy bank." If we are not simply arguing past each other, we must be understanding the same thing when we talk about this 'hundred-dollar bill'. If existence could explicate the concept 'hundred-dollar bill', on the other hand, one of us would be talking about an existing hundred-dollar bill, and the other would be talking about a nonexisting hundred-dollar bill, and we would not be using the same concept. We would be talking about different things entirely. As Kant puts it, "it would not be exactly the same thing that exists, but something more than we had thought in the concept; and we could not, therefore, say that the exact object of my concept exists." So, says Kant, existence is not part of the content of a concept; it is, instead, simply a judgment: in an ordinary judgment, like the one about triangles above, we propose that what fits my concept has a feature determined in the predicate, while in an existential judgment, we propose that there is an object for my concept. And to have grounds for doing this, we have to appeal to experience. This is the kind of thing Kant means when he talks about existence not being a predicate.
Incidentally, Byrne's division of the arguments is very bizarre. It's not a natural one, since it isn't based on any formal or thematic characteristic of the arguments; moreover, many types of arguments simply drop out as not fitting very well in either category (and, in fact, despite his sweeping conclusions, Byrne doesn't consider very many).