Re-read the Nitrate test specifically and follow the directions exactly. There is a step in there I don't think you're doing. I don't remember it exactly off the top of my head but its like this:
Shake bottles. Fill tube up. Drop bottle #1 however many times it says. Cork tube. Shake tube to mix up bottle #1 and water for time listed. THEN you drop in bottle #2.
I don't remember you mentioning the shaking between bottle#1 and bottle#2. It also says to shake bottle #2 for a full minute before using it.
I've only used the test once so far as I'm new to getting my own aquarium, but I remember that much.
Also: As for your tap water readings: If you run the water for a few minutes before testing you might find different results, and since you've been using so much water lately you may have flushed out alot of the bad things from your plumbing system and are now getting a bit fresher/less metal/ammonia'd water perhaps.
Anyways, my first post and I hope it goes well!