I have a system were both the signal generated and the sensing equipment is sharing the same ground.
Which one of these shield terminations scenarios is correct? (if any). I searched the internet for about 2 hours and could not find a clear definite answer.
None are correct if you want accuracy and the cable is long. You have to consider the fact the although both grounds are supposedly connected there may be ground currents (due to other equipment) that will create milli-volt differences along the ground and this adds to the signal received. This means you get signal+stuff feeding into your sensing circuit.
You also have to consider air-borne noise and to properly cancel this you need impedance matching at the sending end and a differential amplifier at the receiving end. The noise that "hits" the shielded cable may induce currents in it and these currents will also induce currents in the central "hot" wire. If those currents do not terminate to ground in the same way you get a difference voltage and a noise voltage at the receiver.
Of course, if the cable is short and you have the "system" in a "clean" environment then you won't see much noise or interference.