Almost,
The bacteria which turns ammonia into nitrite is Nitrosomonas.
The bacteria which turns nitrite into nitrate is Nitrospira. It was thought for many years to be Nitrobacter, but Timothy Hovanec proved that this was in fact not the case. Nitrobacter is a genuine NOB, but not the one predominant in an aquarium environment.
I can point you towards his papers on the subject if you're interested.
Oxidation is the name of the process of turning ammonia to nitrite, and nitrite to nitrate.
When Hovanec discovered that the correct NOB was Nitrospira, he patented the idea of bacteria in a bottle containing nitrospira, so nobody can make it without his permission. He was working for Marineland at the time and they made Bio-Spira, which no doubt you will have heard of? It was really quite good and basically the only one which actually worked.
I'm not sure what happened between them, but he doesn't work there any more and Bio-Spira is no longer the product it used to be. I'm fairly sure that this is because he took out the patent in his own name, not the company name, so now Marineland can't make the product which they once did.
He does have his own products out now called Doc Tim's The One and Only (or something similar), but I've never heard of anyone who has used it. I'd be interested to find out what its like.