z represents frequency differences which is the same as apparent slowdown (or speedup for blueshift). Note, this is apparent slowdown in the sense of what we see through a telescope, not how much it is "really" slowed down in "our" reference frame.

Now, when we imagine an extending our reference frame to that other galaxy in a particular way such that in that particular extension of our reference frame the slowdown is caused by motion rather than by universe expansion, then we use the relativistic doppler shift formula to get a speed. That formula involves a Lorentz factor (or rather a sqrt ((1+v/c)/(1-v/c)) factor).

Edit: for clarity, the relativistic doppler formula I think should be better represented as (1+v/c)/sqrt(1-(v/c)^2). This makes it more clear that it's a Lorentz factor (the denominator) representing the relativistic time dilation in combination with the numerator which represents the non-relativistic doppler effect (due to the time it takes light to get here increasing as the thing moves farther away).

Another later edit: We actually don't want to just use a doppler formula, at least if in the standard picture the expansion rate of the universe is changing. That's because the expansion rate changes via a gravitational effect that would also be expected to have a gravitational doppler effect. So in a no-expansion picture we want a combination of doppler effect and gravitational redshift (at least for a changing expansion rate), just nothing from stretching of space.