According to Stryer there are four major groups of photosynthetic prokaryotes, i.e. Table 19.1
Bacteria Photosynthetic electron donor $\ce{O2}$ use.
Green sulfur $\ce{H2}$, $\ce{H2S}$, $\ce{S}$ Anoxygenic.
Green nonsulfur Variety of amino acids and organic acids Anoxygenic.
Purple sulfur $\ce{H2}$, $\ce{H2S}$, $\ce{S}$ Anoxygenic.
Purple nonsulfur Usually organic molecules Anoxygenic.
Cyanobacteria $\ce{H2O}$ Oxygenic.
Do each groups belong to a different genus?
To answer your question, yes they are all in a different genus, but that’s really specific. If you are wondering about their fundamental differences, they fall into a few different groups.
Green sulfur bacteria usually falls into the phylum Chlorobi
Green non-sulfur bacteria falls into the phylum Chloroflexi (Part of the terrabacteria superphylum)
Cyanobacteria is a phylum in itself but also falls into the terrabacteria superphylum.
Both, Purple sulfur/non-sulfur bacteria are all proteobacteria.
Now, pay attention here, as linking these could be somewhat confusing sometimes. But now we can link together their relationships.
Green sulfur bacteria (Chlorobi) falls into the FCB superphylum, meaning that it is a gracilicute.
Purple sulfur/non-sulfur bacteria (Proteobacteria) are also gracilicutes.
Green non-sulfur bacteria (Chloroflexi) is part of the terrabacteria superphylum. Terrabacteria are not gracilicutes.
Cyanobacteria is also in the terrabacteria superphylum. (Again, not a gracilicute.)
Now that we know this, we can reasonably assume that both purple sulfur/non-sulfur bacteria are related to green sulfur bacteria, as they are both gracilicutes. Also, we can conclude that the non-gracilicute green non-sulfur bacteria and Cyanobacteria are more closely related to each other than they are to the latter mentioned bacteria.
Here’s another cool fact, some Deinococcus-Thermus bacteria (members of the terrabacteria superphylum along with Cyanobacteria and Chloroflexi) have been found to carry the bacteriophytochrome photoreceptor (BphP) which absorbs red light. Interestingly, Deinococcus-Thermus are non-photosynthetic, but these photoreceptors are found in photosynthetic Chloroflexi as well. This indicates that the deinococcus bacteria likely diverged from chloroflexi at some point but still retained the BphP.
Hope this helped!
Genus is the classification one more general up from species. Two species in the same genus will be pretty similar - for example wolves and coyotes. Two things with radically different energy metabolism systems will not be in the same genus. In fact these different modes of metabolism mean that the creatures using them are so different from each other that they are categorized in different phyla (phylum being the second division after kingdom). Humans are in phylum Chordata with frogs and fish. Prokaryotes using such different modes of energy metabolism will not be closely related at all.
