If you believe in second-order logic then you believe there's only one second-order logic.

I'm not sure whether to interpret that as a novel form of other-optimization, or as an ironic take on the idea that if one believes in arithmetic ( or set theory, for that matter) one also believes that the subject matter is unique.

In any case, my personal favorite higher order logic is the internal language of the free topos, which is, in fact, unique up to isomorphism. But far from universally accepted.