When I was much younger I really had trouble tuning the drums. But now after I came back to drumming after a career in mechanical engineering I think I understand it on a different level now. I am not saying I am expert at all, I just have a different insight to tuning now (End of disclaimer).
If a head/ diaphragm moves in a uniform pattern that is symmetrical and about the center, up and down like a speaker movement, you will have resonance. If the head/ diaphragm moves in a wobbling uneven pattern from side to side, and not from the center you will have the ringing. A thinner head will be easier to move from side to side because it does not have the structure to resist the movement, and it takes less energy to move in the side to side pattern. Also if you tune the head very high, it would be more difficult to let the head move from the center, because it takes more energy then the side to side pattern.
The heavier head will be able to resist the movements from side to side better. Adding tape, mufflers, or gel to the head adds its resistance to move in that side to side pattern. If you do not have the heads tuned evenly it too will also cause the side to side pattern of the diaphragm movement. If you tune the resonant heads loose, it will not ring as much, because now the heads can move from the center easier. A lot of drums have trouble tuning the resonant heads loose because the lugs fall out. Many drummers use lock tight, rubber washers, or tight-screw to keep the lugs from falling out.
A denser wood will allow more energy from the batter head to be transmitted to the resonant head. It takes a lot more energy to move the head in the symmetrical pattern then the side to side pattern. Simply because there is more mass in the head to move from the center then the sides.
In conclusion the ring is the side to side movement of the head/ diaphragm. You want to eliminate that side to side movement by, thicker heads, loosely tuned resonant heads, or mufflers to the heads, and evenly tuned heads.