What I think I remember from my days as an active birder (long past, alas, long past!): Here in the Washington D.C. area the Yellow-Billed Cuckoo is one of the last of the migrating songbirds to come through - late May/early June? If it is similarly late farther along its migration route, that might have given rise to the notion that it "never sings cuckoo 'til the fourth day of July."

Or, might the several versions of the song be of European origin?

I am not certain that the Yellow-Billed Cuckoo is parasitic - i.e. lays its eggs in other birds' nests. Around here, it is the Brown-Headed Cowbird that exhibits that behavior. I assume that at least some European cuckoos are parasitic.