Another way of expressing the difference between first order and second order languages?

It seems to me that in a language with logical constants (including
quantifiers and sentential connectives), variables, and nonlogical
constants, the language is first order if it allows the quantifiers to
apply only to variables or to nonlogical constants but not to both, and
the language is second order if it allows the quantifiers to apply to
both variables and to nonlogical constants. Is this a correct analysis?
I've seen the analysis that applying quantifiers only to variables
results in a first order language, and that applying quantifiers to
both variables and nonlogical constants results in a higher order
language, but I've not seen any analysis of a language which allows
quantifiers to apply only to nonlogical constants but not to variables.
But I suspect that actually my hypothetical language is actually the
same thing as a standard first order language, with effectively just
the definitions of "variable" and "nonlogical constant" swapped. Hence
my question.

This leads to the further question which I posted here in another
thread a few minutes ago (message
<1129593284.845677.225340@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>) about whether
the labeling of a group of symbols as "variables" instead of
"nonlogical constants", and vice versa, is really not formally
determined but instead is just a convention about the kinds of
interpretations normally applied to those groups of symbols, with the
only formal distinction being that there are two separate groups of
symbols to which models can be applied independently.

.

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