Interpreting Phillips-Perron Unit Root Tests

I would like to make sure that I'm properly understanding how to interpret the Phillips-Perron Unit Root Tests available in SAS. The best resource that I've been able to find was this page: https://support.sas.com/resources/papers/proceedings15/3294-2015.pdf

Example test results:

Phillips-Perron Unit Root Tests
Type        Lags    Rho         Pr < Rho    Tau     Pr < Tau
Zero Mean   0       -1.973      0.3281      -1.02   0.2703
Single Mean 0       -22.7264    0.0013      -4.17   0.0027
Trend       0       -22.7963    0.0105      -4.06   0.0174


My understanding is that I should look only at the Pr < Tau column, ignoring the Pr < Rho column.

I should look at the types in the order:

1. Trend
2. Single Mean
3. Zero Mean

If I can reject the null hypothesis at Pr < Tau in the Trend row, then I don't need to bother looking at Single Mean or Zero Mean. In the provided example, this is the case with .0174 being less than .05.

Rejecting the Trend hypothesis indicates that there is a high probability that there is no unit root and that the process is trend stationary.

Rejecting the Single Mean hypothesis indicates that there is a high probability that there is no unit root and that the process is stationary, but the mean is offset by some constant from 0.

Rejecting the Zero Mean hypothesis indicates that there is a high probability that there is no unit root and that the process is stationary with the mean being 0.

Did I say anything that is false or shows a superficial understanding? Is there anything I'm missing?