I think almost all 18650 cells have a fuse in there. If you burn the fuse, the cell is dead. Some sellers might call this fuse "protection."
I have used polymer batteries with an internal circuit that cuts out at overcurrent and have to be reset. That's annoying, because you have to put 4V on it to wake it up, again, and the protection acts like a fast fuse, and it should really be a slow-blow, IMO. I haven't seen that yet, on an 18650, but apparently they exist (^ and it adds 3-4 mm). Yeah, you will lose a tiny bit of power to the internal FET, but that will be very little. I wouldn't worry about it.
All li ion chargers regulate charge current and max voltage. The circuit you linked also prevents overdischarge. The protection on the cell itself is often just for max current (for discharge. I dunno if it also works for charging, but the charger should take care of that). If your cell has extra goodies on the protection, then it might be redundant. That would be unusual, though, to have undervoltage protection on a cell, itself. The cutout voltage depends on the load, so it's not typical to add that to a generic cell. The board you linked decided that for you, but the optimal cutout will depend on the draw at the time, and is typically up to the designer/engineer to decide where that should cutout.