Yes, there will be some axial movement with the part you circled in red in the second photo in post 29. That motion relates to the opening and closing of the PMV. Although you will feel a lot of axial movement, in actuallity it only moves a very tiny amount when the system is at operating pressure.
I have included a photo of the assembly order of the cap.
The copper washer goes into the cap, then the spring, then the brass piece goes into the ID of the spring. There is a depression in one side of the brass piece that sits on top of the pin in the top portion of the PMV. Being you have to assemble it sort of inverted, you can put a dab of grease on the brass piece where it inserts into spring to hold it in place. Same goes for the copper washer to either hold it in the cap or on the top of the spring. Then tighten the cap to the desired pressure or dimension you previously measured before disassembling.
Looking down into the top of the PMV the top of the pin should be relatively flat. The depression in the brass piece sits on that pin.
The last two photos are of my PMV. Your PMV has to lines coming out of it, mine only has one, might be because you have auto shut down.
I sent you a PM.