Based on the descriptions, the left side shows what is probably inside these receivers.
The first diagram that uses the "NPN" is safer than the "PNP" IMO.
Yes, you need to connect the ground to the RPi.
Edit: added a third option, most conservative.
simulate this circuit – Schematic created using CircuitLab
First of all, sensor TX pin is transmit output so it must be connected to RasPi RX pin which is a receive input - Connecting TX to TX is usually wrong. But in this case is not that simple, so don't do that just yet.
The sensor TX output voltage levels are same as it's supply voltage, so it uses 5V logic levels. RasPi RX pin does not tolerate 5V logic levels,...
The DB9 breakout board that you specified is also a level-shifter. Supply it with 3.3V from the RPi. Then you can connect it directly to the RPi GPIO pins, no resistor is necessary.
The hardest part is getting the signal direction correct (3.3V and RS-232). TX & RX are only meaningful when the point of view is known.
The RS-232 DB9 cable may need to be ...