Your doing fine. A fish in cycle is tough, I did it that way too. No one at the FS ever mentioned cycling a tank. I got the standard 24 hours and your ready crap. So don't feel alone in not knowing the right way, MANY of us have done it that way, to many.
My tap water also test + for ammonia so I don't worry unless it goes over the .25ppm.
Your tank is in the nitrIte phase of the cycle and nitrItes can be more toxic than ammonia. So now it's even more important to test your water twice a day for that pesky nitrIte and do as many PWC as it takes in a day to keep the nitrItes below .25ppm, even a 90% PWC if necessary.
GOOD LUCK and YOUR DOING JUST FINE