My Sideline quarterbacking for the OP...
The Good....
Looked to me like you kept the speed up, this allowed for good aileron control when the turbulence was encountered.
I couldn't tell for sure but it looked to me like you were not quite full flap. Power wasn't required, but if it had been it would have been much more effective with less flap
The pilots control response was immediate, minimizing the upset
737 wake turbulence is one of the milder ones compared to some aircraft (like a 757), what you hit was about the worst I would have expected in your configuration and situation.
The Not So Good.
They warn you about Wake turbulence and we teach about it for a reason, I agree with some other posters that there should have been some better risk management and recognition on the potential risks. I would have liked to heard the pilot talk about it a bit. How I responded as a CFI would have depended on the OP's experience level. In teaching mode we would have talked risks and mitigation and I would have recommended corrective actions. In evaluation mode I would have probably been quiet and just made sure we didn't get into a corner I wasn't sure I could get out of, ie. get low, slow with full flaps prior to likely hitting the turbulence. Letting you hit some turbulence is something you will remember a lot longer than me just keeping you out it.
Power setting didn't appear to be to bad (1500RPM) about the max I use for a normal approach. But a lower power steeper approach would have like kept you out of the wake turbulence.
IMO most Single engine VFR aircraft should be approaching on the High Side of the VASI/PAPI lights. I couldn't seen any Visual approach indicators in the Video
Getting lower over the approach end of the runway after a wake turbulence warning. It actually appeared you were a bit fast and high energy and the floated down the runway a ways, you brought it down to the runway early instead of staying on a bit higher approach path to a point past the touchdown point of the 737. From the conversation it appears you were trying to get it in shorter. Not really a good thing when dealing with potential wake turbulence
Summary: it was a pretty minor wake turbulence encounter because you weren't excessively slow, reacted immediately, and it wasn't a severe hit of wake turbulence.
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A couple comments on the 2nd video.
Dash-8's are pretty mild wake turbulence as well, But it became a big issue because of the low and slow approach along with the students slow or no reaction to the encounter. Full flaps may have made the recovery slower and more challenging but i don't know what the flap setting was.
Brian
CFIIG/ASE