It is only a question of how you wish to use the term "exist." It's just a word, it can be used any way you like.
- You may use the word "exist" (or "real") to refer only to physically existent things in our current physical universe, such as hamburgers but not unicorns.
- You may use these words to refer to things that could conceivably exist, even if physically they do not exist in this universe.
In your quote about the "possibilist's thesis," the "possibilist" uses the word "are" (as in "there are") to refer to things that might not physically exist, as long as it is conceivable for them to exist. He uses the word "exist" to refer to things that physically exist in this universe.
There is not a clear consensus on how to use these words, but here's what I'd suggest:
- Usually, use "exist" in the strict sense, to talk about what physically exists in this universe.
- When speaking of mathematical objects, we may use the word "exists" in a conceptual sense, but it should be understood that this is a different sense than the sense in which objects exist physically. It might as well be a different word.
- When speaking of fictional objects, we may also use "exists" in a conceptual sense. If we are asked whether they "really exist," there are two answers: within Superman's fictional universe, yes, Superman "exists" and "really exists." Within our universe he neither "exists" nor "really exists." It simply depends on whether you are using the "physically real" or the "conceivable" sense of the word "exists," which depends on the context.
So, to sum up, my suggestion would be to use "exists" by default in reference only to physical existence in this universe, but to switch to other senses of the word as clear from context, without forgetting that they are different senses.