I think, this would be the correct direction to dig into.
Sun light is very powerful. A comparable small ray, reflected by a mirror could easily outnumber a powerful laser. This ray could be so bright, that even light scattered from this ray, could produce these flashes shown in OP.
I'm not at a final conclusion, but some hints:
The flashes observed appear far larger than the mirror.
In fact, when the mirror and camera were down close to the water, the flashes were strange odd shapes.
There are 3 shapes we could expect:
A point: Perspective and distance could make it look like a point.
A circle: An out of focus point.
A rectangle: The shape of the mirror.
(And when they went up above the water, it did seem to take on a more regular shape)
But down by the water, the shape was often like melted swiss cheese or flying pizza slice.
But it is clear that there was vast distortion near the water, and why shouldn't there be? Look at any vanishing ship and you can see right near the water things are all weird.
Targeting a beam of 3 feet diameter to a camera 13 miles away seems impossible to me. The girl at the beach is tilting the mirror back and forth by several degrees, whereas hitting the camera with the beam would need an accuracy of arc seconds.
If you notice how the mirror shines on the sand at distance, it wasn't flat. It was curved, with a focal length of 50 feet depending on how she was holding it. It was kind of a thin mirror and not too ridged.
This would have caused significant beam spread which would make it a lot easier to get a flash to the camera.
The flashes have significant differences in brightness.
If there would be a direct hit, I would suspect to see a very bright center - brighter than the flashes in the video - and a significant zone of glare around it. These flashes more look like, we only see the glare.
Or it could be a scatter effect, similar to this, what let you see Crepuscular rays.
similar to this, what let you see the light cone of street lights with fog, drizzle, rain or snowfall.
similar to this, when an experimenter in a laboratory makes a laser beam visible with smoke.
If the beam and camera's viewing line are aligned quite nicely, the scatter could sum up to bright flashes, but there's no need for a direct hit. The camera only observes the light scattered by dust/particles/aerosols somewhere in between this 13 miles stretch.