I am just stepping into the teaching world, and finished my first semester as a teaching assistant in an introductory course in set theory and logic, which is giving a formal background to induction, order relations etc. etc..
I can give my own insight as someone who finished his undergrad degree recently. In my university it is mainly the first method for math students. We see proofs, we are given questions which are mostly about proving or disproving things. By the third year I think that everyone in my class (well, we were quite a small class to begin with) knew very well how to write a mathematical proof.
The second approach you described sounds a bit problematic to me, and I'll tell you why. I took a course in functional analysis. Other than the name of a few theorems, and maybe one or two theorems which I actually remember the contents (but not a single proof) I remember pretty much nothing of the course. Same can be said on the course I took in number theory (though I remember slightly more from that one), and on other topics. It's not all bad, when my friend who's taking a related course asks me a question I usually amaze myself by being able to supply a partial answer, and if I ever encounter the material it's easier to go through it. However, I still don't remember much. Giving someone a course in "How to write proofs" means that for some it will stick, and for others it won't stick - and they won't be able to wake up in the middle of the night and give a formal proof to some theorem they will later name "The Dreaming Lemma"; while in contrast it will take a long time for someone who spent three (or more) years just seeing proofs and writing proofs to forget that method, and not to mention the bonus for deep critical thinking which allows you to be able to scratch off ideas even before they reach your mouth or hands.
That been said, I do think that the second approach is very good when you want to focus on teaching mathematics in a lower level (i.e. non-academic level, or even low level math course to philosophy students) or if your students are in applied mathematics program, or something like that. I don't see how many set theorists and logicians will grow from this sort of method, but I might be wrong and even if I am right - not everyone loves set theory.