What does the Catholic Church think about God being defined as "that than which nothing greater can be conceived"?

The IEP article on the ontological argument may be of assistance. I will use it to guide my answer.

There are numerous objections to the ontological argument, some better than others. First, is Gaunilo’s objection. He argues that Anselm's reasoning could be applied to any conceivable genus, and so the greatest possible instance of that genus must exist. He uses the example of a greatest conceivable island:

It is a conceptual truth that a piland is an island than which none greater can be imagined (that is, the greatest possible island that can be imagined).

A piland exists as an idea in the mind.

A piland that exists as an idea in the mind and in reality is greater than a piland that exists only as an idea in the mind.

Thus, if a piland exists only as an idea in the mind, then we can imagine an island that is greater than a piland (that is, a greatest possible island that does exist).

But we cannot imagine an island that is greater than a piland.

Therefore, a piland exists.

The flaw with Gaunilo's argument is that it ignores the special case of "greatest being" generally speaking. The qualities which make a great island are not the same as those which make a great being. A being, most notably, is something which is supposed to exist. So, any being which exists is greater than any being which does not exist. Not so for island, where something like climate might matter. We could imagine a more perfect climate than any real island has, but we cannot imagine a more perfect than any real being, because existing is a great-making property of beings.

Aquinas, however, had a more practical and realistic objection to Anselm's claims. He says: "not everyone who hears this word ‘God’ understands it to signify something than which nothing greater can be thought, seeing that some have believed God to be a body." This means that you can only be convinced that God exists by Anselm if you understand God to be the same thing he does. Aquinas gives an example of misconceptions that real people have about God which may lead them to object to Anselm's argument on the grounds that he gets the definition of God wrong.

Aquinas, instead, posits in his 5 ways that there are basic facts about the world that cannot be denied, which will lead every honest person to conclude there must exist some supreme being, a being that is "first" among all other beings, and has its existence in a way unique from other beings. Indeed, this being is the only one which must exist, of its own nature, and all other beings derive their existence from it. Once this latter point is understood, it becomes evident that this being is not merely "first among" others, but that it is a uniquely different kind of being, and a being which "all men call God." [ST I Q2 A2].

It is Aquinas' approach, and not Anselm's, which modern apologists tend to take. This is not because Anselm is wrong about the nature of God. We can see plainly in Aquinas' conclusion that this uniquely different kind of being must be a being "greater than which no other being can be conceived." Rather, Aquinas' arguments, and derivations thereof, are used because they are far more convincing. The bring the notion of God down from the ideal into the real, and make themselves palatable to ordinary human experience. The first way, for instance, is a demonstration that God must exist because things change from being one way to another. Everyone who is reasonable has to grant this premise: that things in the world change. This is unlike Anselm's premise, which is so lofty as to be unconvincing to those not already convinced (eg, by Aquinas' arguments).

It is therefore not the case that Anselm's argument is "not accepted," but rather many philosophers and theologians agree that, in general, it "doesn't work," at least not on those who are not convinced already that God is what Anselm says He is.