For the sake of discussion, I'm going to suggest why I disagree with Bill (brunerww). My reasons are from personal experience. For over a year I have had a Panasonic camcorder similar in size and shape to the HC-V500 and an HX9V. Both were purchased after reading a lot on this forum and other research. Recently I bought a Panasonic FZ150 "super zoom" for my wife. That was also influenced by discussion on this forum, including opinions from Bill.
First is the question of video quality. Of course there have to be technical differences and skilled or trained pros or prosumers can analyze that. My experience is a little different. Any time I show a 1080p or other AVCHD quality video to anyone on a large flat screen TV they comment with phrases like, "that picture really looks good". (No, they don't always say the video is good!) So, my suggestion is the viewing quality (sometimes called "picture quality) will be very good with any of the cameras considered.
Does Chris need 52x zoom? Not unless the class size, or tournament venue is the size of a football stadium and he is shooting from the cheap seats. Does he need a video light? Not unless martial arts classes are conducted during power outages! Does he need O.I.S (optical image stabilization)? Of course, and these cameras all have that. Both the V500 and the HX30V (or related HX20V) have "improved" O.I.S. Which one is better. Only someone who owns both and tests them will know. The truth is that hand held video shot with minimum zoom will always look more stable.
Chis, as a martial arts instructor, wants to take video while he is working. He is busy teaching! He will get more useful video from a camera that is handy for "grab and go" or "run and gun" shooting.
If I am in the mood to be a "videographer", want to concentrate on it and block out all other distractions I pick my Panasonic conventional camcorder (like the V500). I keep my hand in the strap and often have it on a monopod. Sometimes I use a tripod. I'm "geared up". Because of the shape or "form factor" as a conventional camcorder, it is sometime hard to set it down in a safe secure place. You need a table!
If I am in the mood to join an activity with family or friends and don't want shooting video to be my sole purpose or obsession, I grab my small (and very cabable) Sony HX9V. The very capable camcorder is in my pocket! If there is a potential shot, I pull it out and push the on button and then the shoot button. I could do that with the larger Panasonic too. But, if I'm not already holding on to it, I have to get it out of a case that has to be strapped to my belt or slung over my shoulder. Can the smaller camera by put on a tripod and used like the bigger one? Of course.
To summarize, if Chris intends to make shooting video is primary mission at a tournament or in a class he should get something like Bill (brunerww) suggests. If Chris wants to have video shooting as an accessory or enhancement to his primary focus of teaching, he might do better with a camcorder like one of the Sony's I suggested. The differences are not about the technical features, but the convenience for the mission of the day.