Your API test kit is most likely accurate. Almost all Municipality water supplies show the concentration in ppm of nitrogen as part of nitrate (N-NO3-), which would be lower than if you showed it as ppm of nitrate.
I know it sounds confusing but this is how all water quality standards are reported. Usually showing the concentration of element of interest (N,S,P,K,Fe, or what ever it may be) as part of the entire compound. I do not have my Water Chemistry notes from college on me, and do not remember why they do this. It would explain some of the difference between your API kit and your tap water, not all of it, but some.