I'm away from my library, but fortunately the books you referenced are a Google away, so I could consult them and confirm that they say what you say. I'm not quite willing to accept Kleene's word as an authority on common modern mathematical practice, since he was a theoretical computer scientist before the term, but, though I'm not familiar with Enderton's book, it certainly looks like a reasonably standard one.

But these are all referring to Peano arithmetic as a model of the theory of the natural numbers. And that seems a bit silly: the impact of Peano's work wasn't because he showed that there was a model of the theory of the natural numbers, which everybody believed if they bothered to think about it, but because he showed that all you needed to make such a model was a successor operation satisfying certain axioms. Yes, they may be less model-theoretically congenial because they're second order, but to change Peano's work from what he did historically and still call it Peano's seems strange to me. (I'm fine with dressing it up in modern language, and calling it an initial object in the category of pointed sets with endofunctor, which perhaps is biased but still seems to me to be capturing the essential idea.)

Certainly I was taught the second-order approach, though it was as an undergraduate; I've never taken a model-theory class. As I say, I'm away from my library and so can't consult any other sources to see if they still teach it this way, and anyway I am a representation theorist rather than a logician; but, if the common logical approach these days really is to discard Peano's historical theory and to call by Peano's name something that isn't his work, even if it is more convenient to use, then I think that's a shame from the point of view of appreciating the novelty and ingenuity of his ideas. But just because I think something is a shame doesn't mean it's not true, and so far you've produced evidence for your view and I can't for mine, so I can't argue any further.