The feature you're looking for is call "Focus tracking" or "face tracking" depends on which manufacturer. As far as I know, it's very common on Sony's camera I guess they want to make their camera easy to use. I haven't use Canon point and shoot camera for at least 6 years so I don't know if they have that kind of feature. Panasonic have face/smile recognition, I think the GH4 has the tracking feature too as I remember it's not worth to use.
Here are the example of the feature you're looking for?
Tried it on my dog:
To be honest, I think the feature is useless underwater. First of all, I think the algorithm they designed for the tracking is to look for eye ball because it's the most important focus point of the picture. I believe new camera these days look for not just eyes, but nose, and lips (smile). Fish/turtle/shark etc doesn't have that kind of facial composition so the tracking won't work that well. I remember trying similar feature in the GH4, when the fish swim out of frame, the tracking get lost and I have to constantly re-set the tracking and after a few times, the camera gave error or the focus box just turn yellow and won't turn green to track. It might be useful for close up macro where the subject doesn't move fast and you have bigger eyeballs to focus on, but if you're shooting macro, manual focus with peaking is easier and less fuzzy to deal with.
You can click on my signature to see some videos that I shot with the Panasonic LX100, if you have a retina display, thunderbolt monitor or even a 4K monitor, then you can decide if 4K is important for you. I never use the HFG20, and frankly, the equipment is only half of the equation. The dive destination and skill of the shooter could bring out the capability of the camera/camcorder, even though the equipment is not top of the line.
Ok. Let's say that you see a big turtle in a beautiful spot at the reef. You don't want a take of the Turtle in its environment and after it another one closer just framing the whole turtle, and then a last one of the head of the turtle... You want to shoot a Travelling in by swimming towards the turtle until you ge really close to it.
How do you proceed?. Let's say you need to shoot with maximum aperture and have the focus range really compromised.
What you're asking is called "establish shot", so you show the turtle in 3 sequences: 1.) The turtle and its surrounding, 2.) the whole turtle, 3.) the face of the turtle.
You don't shoot all 3 sequence in one go. First you have to take a custom white balance when you are about 3 meters away from the turtle to get the color correct (if you have wide angle lens, you could be closer to 3 meter). Set focus, then start shooting for about 30 seconds. Stop recording, swim closer like 1.5 meter, take custom white balance (or turn on light), depend on the size of the turtle, you might have to remove the wide angle lens. Set focus, then take another 30 seconds shot. Then swim closer within arm length to the face of the turtle, take white balance/turn on light, extend your arm out, set focus then shoot the face of the turtle.
Once you have that 3 shots, just put them together during post. The reason I shoot 30 seconds or so because the first 5 and last 5 seconds tend to be shaky since I would be pressing the record button.
This is how I would shoot the scene you asked because I only have point and shoot experience. Other people may have different/better way, especially those who use dSLR with lens that have limited focal length.