Hi kris2112 and welcome to the TFF beginners forum!
When you added the mature gravel muck to your new filter there were probably several things of interest that went in. There was organic debris (small bits of plant debris, fish waste etc. from whatever tank it came from), some ammonia, some nitrite, some nitrate and then the gravel with various microorganisms (fungi, single-celled animals, algae etc.) and yes, hopefully some of the 2 species of nitrifying bacteria that are beneficial, stuck all over the gravel in sticky films we call biofilms. The existing nitrates in the muck may have been some of the nitrate(NO3) you measured, along with some new nitrate recently produced by the nitrifying bacteria that came along stuck to the gravel.
To oversimplify, there are 3 substances and 2 little living animals in the "cycling" process you're trying to learn about. First comes the substance ammonia (comes directly off the fish gills when they respire and from their waste, comes also from "heterotrophic" bacteria (different from the ones we are trying to grow!) floating in the water which break down the solid fish waste and plant debris and any excess fish food (which is also organic.)) The ammonia is food for the "A-Bacs" (the ammonia oxidizing bacteria) and for each 1ppm of ammonia they produce 2.7ppm of nitrite(NO2), which is our substance number two and is food for the "N-Bacs" (nitrite oxidizing bacteria, our little living animals number two) which eat it and produce nitrate(NO3) (our substance number three, about 3.6ppm of it) and this nitrate hangs around in our tank either until we take it out with a weekly water change or it gets "de-nitrified" by still different bacteria that may sometimes be present in small amounts (not enough to be significant, thus the eventual water changes in most of our cases.)
Ammonia and nitrite(NO2) are deadly poisons to fish. Now you might ask.. why would mother nature be putting out a deadly poison right at the fish gills (?) and the answer would be, well.. the fish evolved swimming in millions of gallons of fresh surrounding water (!) always flushing away that ammonia! In our little aquariums, the poisons build up quickly. Even small amounts of ammonia cause permanent gill damage and potential death. Even small amounts of nitrite(NO2) cause suffocation, permanent nerve damage and potential death. So a fast way to rid the tank of these toxins was essential. Turns out "mother nature" provided that too (!) in the form of nitrifying bacteria (plants can do it too, but that is not part of our simple discussion here) and ever since the early days of aquariums the answer has come to us in for form of the "miraculous" machine called a "biofilter."
The "biofilter" is just any "system" that allow our two species of nitrifying bacteria, the A-Bacs and N-Bacs discussed above, to stick on "media" (gravel, sponges, ceramic rings, ceramic pebbles, bioballs, lots of things!), grow their biofilm nests and be provided with a constant flow of fresh ammonia and oxygen and water.. that's it! Filters are simple, even crude looking things, but don't be fooled, they are amazing machines and are the heart of the modern aquarium hobby.
It takes about a month or two to grow sufficiently large populations of A-Bacs and N-Bacs for the filter to be ready to be operational and support a fish population. Prior to that their won't be enough bacteria to eat all the toxins and the toxins will show up on our tests (the little "snapshots" we take of water chemistry in our little test tube kits.) The filter hardware and test kits are just raw materials and must be used with the type of knowledge gained here in our forum!
OK, hope that gets you jumpstarted in your learning there!
~~waterdrop~~
(ps. just had to write something because my wife made me do errands just when I was going to post to you earlier and I thought, o darn, now MW will beat me to the case

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