Different Senses of the Quantifers?

Carnapians want to say that different things can be truely said to exist when speaking in different language-frameworks. So the existential quantifier "Ex" will mean different things in these different frameworks. But can there really be multiple different meanings for these different uses of Ex, which would qualify as different kinds of e.g. existential quantification?

An argument that you can't is: The meaning of Ex is determined by its introduction and elimination rules. So any putative kind of existential quantifier would need to obey them. Hence different senses E1 and E2 from different frameworks would both have to obey the standard introduction and elimination rules for Ex. But if E1 and E2 obey these rules, then you can prove E1x from E2x and vice versa. Hence there is no room for ambiguity.

This argument can't be right though, if restricted quantification ('There is nothing in the fridge'. 'All the beers are in the fridge') - something that even the most ardent anti-Carnapians accept- counts as `a kind of' quantification. And intuitively it is. Hence in order to seem like a kind of quantification, a connective need not obey the full introduction rules. It suffices if there's a more limited range of instances of the introduction schema
P(x) --> Ex P(x) that speakers accept, together with all corresponding instances of the elimination schema Ex P(x). (A^B^C..^P(z) > F) ---> F (in cases where z does not occur free in A,B, C... or F). This is what we have for beers in the fridge.

Why can't the Carnapian claim that the same thing goes on with different linguistic frameworks? The different choices for when P(x) --> E2x P(x) is acceptable will each correspond to a different meaning for the existential quantifier. We can even formally represent these different possible senses for existential quantification formally, by saying a kind of existential quantification E_i corresponds to each subset S_i of the set of predicate-expressions (i.e. to each choice of what predicate-expressions the introduction and elimination schema are supposed to hold for).

You are probably worrying that this turns the Carnapian into a kind of maximalist (all the objects in question really exist, different frameworks just correspond to different framework restrictions) but I can't actually see any argument for that. So speak up if you can!