Powering ISO124P Isolation Amplifier
I think I need two different power supplies.


Yes, you need two different (and galvanically isolated) power supplies if you want to maintain proper isolation.


  I am correct? Or should I use the diagram of the Figure 22 or Figure
  23 of the data sheet for powering the ISO amplifier?


Yes, you are correct AND you can use figure 22 or figure 23's recommendations for isolating power supplies using DC to DC converters.

Just treat both sides as needing a power supply: -



And keep both sides galvanically isolated.