Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context.

In ZF one can prove $\not\exists x (\forall y (y\in x)).$ This statement can be read in many ways, such as (1) "there is no set of all set" (2) "the class of all sets is proper (i.e. is not a set)" etc. and I believe that there is a substantial philosophical difference between (1) and (2). The former suggests that the existential quantifier refers to the actual existence of something intended in a platonic way, while the latter interprets $\exists$ as meaning "it is a set". So, in the second case, I would say that the existential quantifier is a way of singling out things that are sets from things that are not sets, rather than a way to claim actual existence of something.

I am a set theorist and I always intended the statement above as (2) because I don't think existential quantification in set theory refers to actual existence. I suspect that also Zermelo intended existential quantifications as a way of singling out sets from things that are not sets, because in its original formulation he introduced "urelements" i.e. objects that are not sets but could be elements of a set. But I am interested in what is the most common interpretation among contemporary set theorists and I have the impression that my colleagues in set theory use (1) more often.

So my question is: from the point of view of someone who believes that existential quantifiers in set theory refer to actual existence, does the statement above mean "the class of all sets does not exist"? Does this interpretation appears anywhere in the literature?

Thank you in advance.