Two initial questions: How is the push rod connected on the cylinder end (circled)? When installing master cylinder does the transmission need to be removed?

 

I've replaced the clutch master cylinder in my '04 Ion, and I'm pretty sure that the push rod is just inserted into a socket on the piston, and is held in place by that molded rubber cap.
The transmission does not need to be removed to replace the master cylinder for the clutch, but it would need to be removed to replace the clutch slave cylinder, which is inside the bell housing of the transmission. Getting to the master cylinder is much easier if you remove the electrical box (relays/fuses, etc) in order to gain access.

 

Turns out there’s a tiny ring with a few prongs that holds it in place. One of the prongs broke allowing the rod to slip right out. My new favorite mechanic simply replaced the clip from a donor cylinder instead of replacing the whole part, bleeding the clutch etc. Cheapest shop bill I’ve ever had!