The time period calculated as per the Rayleigh method is printed as part of the analysis output file as shown next
You are correct. The table 12.8-2 would yield the same value for X and Z. So Ta would be the same but T calculated as per Rayleigh method would be different for X and Z directions. For example for your model the T for the Z, that was calculated using Rayleigh method, came out to be 5.282 sec. However none of the time periods T calculated by Rayleigh method is governing in your case as these are greater than Cu x Ta. So eventually time period used is coming out be the same. If you want, you can always override the time period calculation by directly defining the periods yourself in each direction ( through PX and PZ within the parameter definitions )
STAAD follows the ASCE and not AASHTO code.