Al-Farabi claimed that existence is not a predicate, because "exists" as in "Apple is red and exists." doesn't bring any new information, but does a predicate have to bring in a new information in order to be a predicate? I thought a predicate only has to be able to be either false or true, so is Al-Farabi wrong?
Rescher has an informative discussion of al-Farabi's views on existence as a predicate in A Ninth-Century Arabic Logician on: Is Existence a Predicate?, which includes their comparison to the more familiar Kant's views. They are quite nuanced and surprisingly in tune with modern approaches. He does not declare, like Kant, that existence is not a predicate, but explains that it is and it isn't, in two different senses. And clearly describes both "naturalist's" (Fregean) and "logician's" (Meinongian) senses that we have represented in modern discourse, see What are the counterexamples to Kant's argument that existence is not a predicate? for a review. It is not that one conception is "right" and the other "wrong", they just deal with different senses of "existence". But it is remarkable that al-Farabi, back in his time, was able to clearly distinguish both.
The naive conception introduced at the beginning of introductory modern logic texts is that a predicate is simply a description with definitive truth conditions. In other words, there is a procedure that delivers unequivocal yes or no on any legitimate object it is applied to (at least assuming God's abilities, we may never be able to tell due to cognitive limitations). The legitimate objects can be abstractions, like numbers and sets, or physical objects. In this nominal sense existence is surely a predicate.
However, as we get deeper into the semantics of predicate calculus things turn more subtle. The problem is with what kind of "procedure" is acceptable. The idea is that we have object language with variables that refer to objects in a pre-specified domain of discourse, and predicates that refer to their properties (and relations, but let's focus on one-place predicates). We must be able to first assign our predicates to a variable, this will be our concept of the object, and then go look for an object in the domain that matches our concept, if any. We pick individual objects one by one, apply the truth conditions from the predicates, and deliver the verdict, yes or no.
We cannot do this with existence. After the concept is formed we can go survey the whole domain to see if it is realized there. Only at the end of this survey can we can say it exists, if there was at least one yes. We cannot add it to the concept before the surveying, and then query individual objects on whether they have it. It is a property of the concept (second order property, meta-property), not of an object. Existence "does not add a thing to the concept of a thing", as Kant put it, hence it is not a real predicate. Thus, in the standard predicate calculus it is expressed via the existential quantifier ∃, which applies to concepts (combinations of predicates), as in ∃xP(x), not to individual variables.
We find this exact distinction between nominal and real predicates in al-Farabi, expressed much more clearly than Kant later did. Kant's quip above echoes al-Farabi's "a predicate must furnish information about what exists", but al-Farabi restricts it to naturalist's conception of a predicate. Here is Rescher's translation of al-Farabi's text:
"Question: Does the proposition "Man exists" have a predicate, or not?
Answer: This is a problem on which both the ancients and the moderns disagree; some say that this sentence has no predicate, and some say that it has a predicate. To my mind, both of these judgments are in a way correct, each in its own way. This is so because when a natural scientist who investigates perishable things considers this sentence (and similar ones) it has no predicate, for the existence of a thing is nothing other than the thing itself, and [for the scientist] a predicate must furnish information about what exists and what is excluded from being. Regarded from this point of view, this proposition does not have a predicate. But when a logician investigates this proposition, he will treat it as composed of two expressions, each forming part of it, and it [i.e., the composite proposition] is liable to truth and falsehood. And so it does have a predicate from this point of view. Therefore the assertions are both together correct, but each of them only in a certain way."
The "real" semantic conception became the canon after Frege and Russell adopted it. There is, however, an alternative modern conception that goes back to Meinong, and that al-Farabi also anticipated in the above passage. It allows non-existent objects into the domain of discourse. The domain now is not "reality itself", as envisioned by Kant, Frege and Russell, but another linguistic layer in addition to the object language. We then split the sense of existence in two, being (in the domain) and existence proper (in reality). The former is a property of concepts and is expressed by ∃ as before, but the latter is a first order property expressed by the existence predicate E! applicable to individual objects. Bucefalus, Alexander's steed, has it, but Pegasus, the flying horse, does not. The expression ∃x¬E!(x) (there is a non-existent object) then becomes perfectly intelligible. And this is arguably closer to the introductory naive conception that is typically fuzzy on the nature of the domain of discourse and the sense of "existence".
Existence is a very broad subject in philosophy with lots of schools of thought. The traditional school of thought, shared, probably, by the large majority of philosophers is that existence is something that an object has. That is, to say that an object exists is to attribute some feature to the object. We might call this the metaphysical concept of existence.
What you seem to have stumbled over is that people who believe in a metaphysical concept of existence differ over whether it is a property or some other sort of attribute. What sort of attributes might there be that are not properties? Well, here you get into some very intricate metaphysics of the sort that can't really be done justice in a paragraph or two, however, one argument goes something like this: existence is a precondition for any object to have any properties. Since existence is a precondition for properties, it cannot itself be a property or it would have to be a precondition for itself.
There is an alternative to the metaphysical concept of existence: the logical concept of existence proposed by Gottlob Frege who claimed that existence is a sort of second-order property. That is, to say that "ghosts exist" is to say that there is an X such that X is a ghost. The existence then belongs not to individual ghosts, but to the class "ghost".