Alright, I'll bite with some nitpicks:
I'll preface this with: Your technique is overall A-OK. I'd be happy handing you my frames to string. You can speed up your mains slightly by hanging on to the string end. Generally speaking, you'll be stringing at max 3 ahead on any given half of the mains. This means at times you'll be doing up to six thread-tension-clamps on any given side. This implies using the same string six times. You drop the string every single time, then spend a few seconds finding the string end. Keep the string end rolled up in a pinky/ring finger while you're tensioning and manipulating your clamps. Once you've clamped, you've already got the string end in your hand, next is just to find the next grommet you're working on. This will save maybe 30 seconds or so (minor, right? Well, not over the course of a pile of 10 frames).
Second gripe is: You tension the "one behind" string before pulling the crosses through (while leaving the tensioning loop). This is a 'theoretical' benefit. Since the point of weaving one ahead is for the second string to weave 'easily,' it should make intuitive sense that it's best to also fan/pull the crosses through as much as possible to minimize friction/string damage. You're also not 'fanning' the crosses much. On a hybrid (i believe) like this one, you won't take any major penalty, but on a sticky poly, you will burn crosses on the top/edges of the frame. You need to be moving your cross more (I see you DO fan it a bit) for safety's sake. This is a major nitpick, I think overall it's going to be fine for the vast majority of strings.
Edit: I should also point out that IF you do change this part of your technique, you lose the "built in" end-of-string finder you have by leaving the last cross un-pulled. IF you try to adopt the above due to string safety, you need to start holding the string end, or else it'll actually slow you down
I think your weaving technique as it is (half stich half push) is actually pretty AWESOME. That stitch weave will benefit you on the bottom 1/3 or 1/4 of the frame. It's hard to push weave with very little room to "V" weave. I wouldn't change it at all. The variety will help, and I imagine polys are actually easier for you on average due to the stitch. It's also very low impact over many many frames, I'd imagine.
Also... are you using an awl as a setting off tool?
Move those clamps out from under the frame when you're doing it
I'd say your level of stringing is pretty high. Don't bother trying to push weave at the beginning, your stitch is quicker than you think. Starting above/below doesn't matter, as long as you're consistent. Developing and having a habit is a good thing, IMHO. I start under because I find it's easier to guide the cross into the hole when it ends over rather than pulling slack and then poking it into a grommet. It's marginally faster, as above.
If you have absolutely no preference, this isn't a big deal, but I'd argue having a "style" here is a good thing. If you always start under (on the outside most cross), you have an error checking method (or misweaving check). If you always expect to start under the outermost cross, and you find yourself starting over (due to the previous string), you've got a mis-weave check built in to your process. If you always start under, and the previous string ENDED under, you know you've made at least an odd number of mistakes on the previous cross (skipped one weave or three). If that makes any sense... Speed wise, I don't think it matters unless you find guiding the string straight into the grommet easier one way or the other. It's one of those "diminishing returns" issues, but that'll get you your 30s-1m shaved off in the long run.
I don't generally string strictly for speed, but eliminating wasted movement adds up over the course of a large pile of racquets, which is why I even think about these things to begin with.