Definitions are chosen, and associated with names, according to their
utility in the development of the mathematical theory at
hand. Sometimes they are defined very locally for a specific proof,
but remain uninteresting outside that proof. Other are given a wider
scope because they can be more widely useful.
There may be some aesthetics in the choice of definition, when it does
reflect some deep structure of the problems, and that will contribute
to usefulness. But a symmetry that is seldom encountered is probably
not that useful, or indicative of deep mathematical structure.
I can only talk of my very limited experience. Closure under
intersection with regular sets is almost ubiquitous for families of
languges. No counter-example comes to my mind. Of course, it is always
possible to contrive one, but not so easily (I guess) to contrive one
that corresponds to a useful family of languages.
When you choose a definition, thus focussing you interest and your
work on it, you have to ask yourself two questions:
How often will it apply to problems you may be interested in, and
how inportant these problems are likely to be (though that may be
hard to estimate)
How convenient is this definition is to derive general results that
can be applied to whatever will fit the definition.
Trios and AFLs where defined as they have been by people who found
that many families of languages met the corresponding definitions.
Furthermore, from the definitions, there were able to prove a variety
of results abstractedly for trios and AFLs, that can be applied
without further effort to all these families.
They apparently had no incentive to do it for arbitrary intersection
within the family. Hence no definition and no name, that I know of.
A further remark is that all these families include regular
sets. Hence they are more likely to be closed under intersection with
regular sets than under arbitrary intersection within the family
(which does imply closure under intersection with regular sets). Hence
the definition as chosen is more likely to be applicable, hence more
generally useful. And it turn out to be already very useful by itself as remarked by Jan Hendrik.
The following is pure speculation on my part. It may also be that
operations considered for closure correspond to interesting structural
properties of languages. For example closure under union seems
naturally associated with non-determinism (recall that deterministic
context-free languages are not closed under union). It may be that
there is no simple such property associated with intersection. But as
I said, this is not based on hard understanding.