What you seek is a CAMERA that has a socket to connect a separate foot switch for on / off operation of the recording process. I don't know of one, but surely some must exist. The item linked above is the foot switch sold as a $100 accessory for a special-purpose CAMERA that you are looking for. That medical equipment website sells the related camera for $2500.
What may be more practical for you is editing software. You use any camera that can record video under manual control by button on the camera AND can transfer the recorded info into a computer. Then you use the editing software to cut out the stray stuff.