Choosing inverse Z-transform equation, given that $|a|<1$
Which one you choose is quite arbitrary: it depends on which region of convergence (ROC) you choose.

As Example 2 and Example 3 of that wikipedia page shows, if you want the signal to be causal, then choose the first one. If you want the signal to be anti-causal, choose the second.

Another way to think about it is what happens when $|z| = 1$. On the unit circle is where the Fourier transform of the signal is defined. If you want that to be finite, then you need to choose the inverse that includes the unit circle.