I dont know exactly what kind of filtration and what kind of resins/media is used to soften the water thus cant say for sure but at least you dont have NH3 now.
I dont see a connection b/n NO2 to NH3 (although reduction reaction/process is possible but we would see such reaction/process taking place in our aquarium if it were true therefore NO connection in reduction process/reaction of No2 to NH3 at least by biologically,IMO).
My first guess is either run the water for awhile before /collecting/testing (remove water that's been sitting in the pipe/filter) OR
You could have containerful of filtered tap water circulating at all time (RESERVE) in large plastic container w/
powerhead (PH) + mechanical media (Hagen's Aquaclear PH has such mechanical media called Quick Filter). You will be surprise to see media turn brownish and slimy in no time (YUK). Imagine drinking water which passes such yucky stuff, especially water which have been sitting in such media.
Anyway, when such filter is ran in the containerful of filtered water, it should go bio-active in time to reduce Nitrite, especially if water has source of NH3 from tap, even with no fish,
OR you could simply use product such as PRIME.
First two suggestions are cuz you seemed to be baffled by such results and by doing couple of experiments, you maybe able to find answers on your own. SATIFACTION of Accomplishment!, LOL!
Which ever route you choose to go, hope all goes well!
BTW, small (20-30%) amt of water replaced during PWC should pose no problem as long as level are low as it is.