I think as a rule of thumb, if you have to mentally keep track of some internal data structure to make sense of your invocation, you're usually pretty deep in "imperative" territory.
> and recursively call “do_things_this_way” on one branch and “do_things_that_way” on the other.
That's the usual algorithm by which imperative logic is translated into equivalent functional logic though.
We know for about as long as computers exit that the two are equivalent in terms of computational power - in the sense that any imperative program can be translated into a functional program that does the same thing and vice versa. That doesn't mean the two programs are identical though.