Plants and bacteria compete for ammonia. This is true. But the plants want it in the form of ammonium (NH4) and the bacteria want it as ammonia (NH3). The big difference between the bacteria and Archaea which use ammonia and the plants is what they out put. Some bacteria and the Archaea create nitrite. And that, in turn is converted to nitrate. The odd part of this is the bacteria which convert nitrite to nitrate are also able to convert ammonia straight to nitrate. In either case, the result of the microorganisms is normally nitrate.
Now the plants use the ammonium but they do not make either nitrite or nitrate. They do a lot of things but the one we most often see in a tank would be oxygen. If the plants cannot get enough ammonium, they will use nitrate if it is available.
So the way to understand the cycle in an aquarium with live plants is that the plants are a part of the cycle. bear in mind for us the cycle is what keeps the more toxic forms of nitrogen, ammonia and nitrite, from being able to harm or kill our fish etc.
There are two more important facts to understand which puts all of this into perspective. We can have a tank with absolutely no live plants that is cycled. This job will be handled by two things. The most important and major contributors are the microorganisms. However, if we also have algae, then they will be using some of the ammonia.
The second fact is that ammonia in water is mostly NH4. This is way less toxic than NH3. Our test kits pretty much all measure Total Ammonia which is NH3 + NH4. pH is the major determinant of how much of the total ammonia reading is in each form. Higher pH = more NH3 and, at 6.0, all the ammonia is pretty much NH4. Fortunately, the bacteria can adapt to using MH4 but do so less efficiently than they can with NH3.
If we want to remove all of the ammonia in any body of water which has a pH a bit over 6.0, we can do so by removing only NH3 or NH4. As soon as one or the other is removed, what remains converts some percent to the other form. In addition to pH, temp. also determine the ratio of the two forms. But temp. is way less important than pH. The warmer the water the more NH3 there will be.
So, the whole point is there can be more to cycling when plants are involved than when it is is just the microorganisms, In fact some plants actually foster the colonization of bacteria in formerly anaerobic areas of the substrate where their roots are. And this causes the development of zones of denitrifying bacteria above and below.
edit to correct typos and tp add "where their roots are" to the last paragraph.