Kids should get the vaccine to prevent spreading it at school or bringing it home. Chances are their symptoms will be much milder than in an adult because they're young and more robust, so their symptoms could be easy to overlook/misdiagnose and get you sick. Its safe, the safety requirements to get a pediatric indication are pretty high.
Scientifically it really shouldn't make a difference which booster you get, they all code for the same crown protein, its the delivery system that's different, which is more significant than it sounds because it affects the rate at which the mRNA reaches the target. For example, the adenovirus has a slightly lower level of efficacy because our immune systems react to the adenovirus vector itself, because we've all had multiple infections over our lives, which is the challenge when using that as a delivery system vs. lipid nano-bubbles that are much less immunogenic. That clinical study referenced is a small study, only 10 sites and 495 people, and they didn't really get into how it was powered, but likely it was for directionality. If you really want to get scientific, have your Dr. perform a before and after antibody titer, its the best way to quantify how the booster affects you. I'm not a Dr., but I lead clinical study teams for a living.