Flooring – What kind of floor tile is this

Pry that tile up, it sounds loose on that corner - i.e. not only unsupported, but some of the thinset in contact with the tile did not adhere.

That looks like unglazed clay tile (aka mexican clay tile), which you must moisten on the bottom to get it to adhere well. If you don't, the tile sucks the water out of the thinset before the thinset can bond, and you get what you have with that "klonking" tile.

If it was just voids at the corners, you could just "pipe" some more thinset under the tile with a pastry bag, but after watching that video, I would pry up that tile. Hopefully you have some spares.

There are numerous floor heating systems that can be used under solid wood flooring, engineered floor or carpet. An internet search of heated flooring will give you links to various floor manufacturers and heating system manufacturers.

You definitely need to check the specs on both the heating system and the finish material to ensure compatibility. There are various restrictions, such as the width of wooden floor boards to minimize problems due to moisiture changes.