Hi there -
I don't mean to burst your bubble, but there are some things about the cycle that you're missing. I will try to explain.
First, you NEED to do a water change. Right now. Those levels of ammonia and nitrite are absolutely lethal to your fish. This won't disturb your cycle, trust me.
You have two kinds of bacteria growing in your filter. The first one turns ammonia into nitrite. That's why at first your nitrite was 0 and now it's higher - that bacteria has had a chance to grow and start making that change.
The second kind of bacteria turns nitrite into nitrAte. That bacteria has NOT had a chance to grow - otherwise you'd have nitrate readings and zero nitrite. So your tank is still cycling.
Taking out a bunch of the water right now will keep your fish healthy. You will still have plenty of ammonia to keep the cycle going - the fish are making more of it every minute, lol.
What you want to see at the end is Ammonia - 0, Nitrite - 0, Nitrate - 10-20. That means the fish produce ammonia, bacteria turns ammonia into nitrite, bacteria turns nitrite into nitrate and you remove nitrate by doing partial water changes every week. Does that make sense?
So my suggestion is this - do a 50% water change right away. After about an hour or two, do another test for all three things. Do this every day until your readings are like I said above - then your tank will be cycled.
It's good that you're doing research, but some people have very different ideas about what is ethical and humane. That other site's method will work, but it's cruel to the fish - they're living in a toilet and are likely to die. This way you'll still get the tank cycled and your fish will be much happier and much more likely to live. I think you'll find that most people here are concerned with the well-being of the fish, and if we sometimes come across rudely, it's because some of us find methods like the one advocated on that site to be offensive.
Hope this helps.