I guess I can agree with what Gary and Lou are saying... Certainly, there's not much to be said for giving geometry students a jumble of betweenness axioms at the start of the course.
But I'm still a bit wary of those who are quick to dismiss those "picky details" as nothing more than that. And textbooks that assume an air of "rigor" without even an acknowledgement of the subtleties that have been bypassed do bright students a disservice.
I don't know what the best approach might be -- I'm sure there are many valid approaches for different students in different circumstances. In any case, at all but the most rigorous level, one can construct plausible, convincing arguments of outright falsehoods. I think one should try to confront students with such arguments sufficiently often to at least make them suspect the existence of levels of reasoning beyond those they are being challenged to master.