What is the theory behind these current equations? (for Z paramaters)
The Io identity applies when I2 is 0.


Just think about how the \$I_1\$ current shares itself out. The 1 ohm resistor is irrelevant and so \$I_1\$ flows into two parallel resistors of which one is 6 ohm and the other is a series connected 4 ohm and 2 ohm. That series resistance is also 6 ohm hence the current shares equally: -

$$I_0 = \dfrac{1}{2} I_1$$


  The reason this confuses me is because I'm doing this with a similar
  circuit which doesn't have that 1ohm resistor, but does have the other
  3, so I need to know if the identity would be different for my
  circuit.


The 1 ohm is irrelevant when connected in series with a current source.

In the second circuit \$I_2\$ (from the voltage source on the right) flows into a 2 ohm load in parallel with a 10 ohm (4 plus 6 in series) load. The far left 1 ohm resistor is now regarded as open circuit because it was in series with a current source.

Can you see now how \$I_0'\$ is formed from your second equation inside the red box?