I would like to know how large or advanced your gym is. I do not think your technique is wrong and mine is right, but I realize that 90% of the people who learn a back handspring are not going to be doing double backs, or even full twisting layouts. To have kids go out and do other things like cheerleading, or just tumbling for their friends, I would have a better peace of mind if I knew they were doing it with straight arms (and therefore safer). By the way I, of course, do not encourage kids to go out and do this stuff in their backyard, but it happens. Also just to clarify, maybe you read my post wrong, but I will never ever teach a BH with hands turned out. Parallel is what I teach. I have seen the effects of a turned-out BH and it's not pretty.
My gym currently has two level 8 girls (well, one L8 and one L8.5) and one level 9 guy (though he's more of a front tumbler), and myself (I'm elite). The experience I draw from in coaching tumbling is not so much experience coaching a lot of high-level tumbling but on what I've found to work with myself, being an elite with floor as my strongest event. Though I have worked with level 9, 10, college, and elite girls on occasion, I've only been coaching for four years, and haven't coached anybody up to that level from the bottom.
You're absolutely right in that most kids never will be doing double-backs. And to clarify, I'll usually aim for hands turned in, but settle for hands parallel. But the natural tendency is to turn the hands out unless a specific effort is made to do otherwise. When my kids learn their backhandsprings (and 9 out of 10 times a kid's first backhandspring is done with hands turned out, regardless of what the coach says), I focus on "getting the hands turned in" to combat that. Sometimes they will end up with their hands actually turned in, sometimes they merely get them parallel. I guess you could say I insist on them being at least parallel before I move on to other corrections and other skills.
I also find that generally the kids who have a strong enough roundoff and good enough upper-body strength can keep their arms straight (or straight enough) even with hands turned in. It may take a little more work, but the difference in how long it takes to learn the BHS is measured in days or weeks; we're not talking several months here.
The hardest part is simply remembering, since often a kid will go right back to doing it with hands turned out if they aren't reminded frequently.
I do not teach very high level gymnasts, but if I did and wanted them to do lower back handsprings, I probably would not mind taking a few weeks re-teaching a single element in tumbling to the 5-10% or so of gymnasts who made it far enough to warrant that kind of tumbling. When a gymnast is that high of a level, a few good drills repeated over the course of two or three weeks I think would be sufficient for them to grasp the skill, especially one so simple, no matter how engrained it is. But like I said, I have not taught many higher level gymnasts and I am speaking from personal experience as a former level 8. You sound better qualified to comment on this.
My experience has always been that these little details get harder and harder to change the longer you wait.
I think this comes down to a difference in methods, and I wouldn't call either way "right" or "wrong."
I have always been taught and teach the arm swing, starting with the arms up, dropping them either straight out or all the way down when you sit, and swinging them upwards/backwards when they jump. Of course you can't do this when doing a running round-off back handspring. Cheerleaders usually start with their arms down and by their side instead of up (in my experience with cheerleading, which isn't a whole lot).
