Yes, the existential quantifier expresses existence.
If you assert that
Some pegasus are flying
then you do assert that pegasuses exist, at least by the classical logical treatment of the existential quantifier, and I would claim also by the intuitive understanding of the sentence. If there are some pegasuses which are flying, then well, there are some ...
According to the authors of forallx quantifiers range over a domain. Furthermore, the domains for classic, first-order order logic are not empty: (page 174)
A domain must have at least one member.
Logic that permits empty domains are called "free logics". As John Nolt describes:
Classical logic requires each singular term to denote an object in the ...
Strawson has worked extensively on this problem of existence of others' minds. He first lays down his criteria of existence:
Strawson lays down the principle that one can ascribe experiences to oneself only if one is prepared to ascribe them to others. This is because a subject can meet this requirement only if they are able to pick out other subjects, ...