How does a capacitive receiver work for a fox and hound set?
The 2nd design is an inductive coupled example using an AM radio. The other two pick up an alternating electric field just like your body picks up AC alternating electric fields as discovered when you touch the input of a speaker amplifier and you get a low hum sound through your speaker accompanied by higher frequencies if you are in the vicinity of other circuits such as switch mode power supplies.

For a simple AC power example, the electric field lines exit a power conductor tangentially and "connect" or "end" at anything that represents an "earth". Normally these field lines are quite short because the neutral wire in AC power wiring represents "earth".

However, you can make a higher frequency oscillator that puts a signal onto a plate/wire and the capacitive connection that the person holding the transmitter makes with ground will cause electric fields to emanate. The receiver, likewise has a plate/probe/wire and can intercept these fields. It has an amplifier to boost the small signal it picks up and this is heard thru the speaker. 

Both transmitter and receiver rely on a person or large object making a capacitive ground connection unless the equipment is powered from AC and earthed already.

That's my take on it anyway.