Splitting nitrification into its two stages: Nitrification happens in two steps carried out by different groups of bacteria.
Stage one, ammonia oxidation to nitrite, is carried out by ammonia-oxidising bacteria such as Nitrosomonas and Nitrosolobus.
Stage two, nitrite oxidation to nitrate, is carried out by a different group including Nitrobacter, so it does not belong with the ammonia oxidisers.
Thiobacillus is unrelated to nitrogen cycling altogether, being a sulphur-oxidising bacterium.
So only Nitrosomonas and Nitrosolobus correctly perform ammonia-to-nitrite oxidation.