I'm a firm believer in the "whatever works" theory. So, for me, if forcing yourself into one style vs. the other isn't really working, the number one "rule" is whatever feels best....
When I miss putts right, it's usually because I don't get my shoulders turned/lined up enough to the pole. I find I have to really exaggerate this in straddle putts.
The other mistake I seem to make most often is not getting my lower body involved enough, like late in rounds when I start to get fatigued. I'll speed up my motion, and it blows the timing and I get nose-down, less "pop" out of the hand, and miss short. I can feel it the second I release. I can't see your vids at work, but I'm guessing that switching from a spin to push style, the lower body becomes more critical to get the consistent speed going.
Cam Todd's advice of going from bent knees to straight (low to high with the lower body,) and from balanced on both feet to balanced on front foot, usually is enough for me to get the "power" behind the putt.
Just spitballing here, but the other thing that's different between the styles is, the longer the putt, the higher my hand has to be on release. Outside of say 25 ft, especially, I make sure my pre-putt routine has me exaggerating a higher release point. This, plus the lower body bit, are what lead to most of my putts that ding off the cage.