What are the criteria that the researchers use to choose whether a protein is an SR protein or and SR-like protein?
Im not 100% certain. If something is called an x-like protein, it usually means it is homologous and has some sequence and / or structural similarity to x. However, there may not be enough biochemical evidence to say definitively that it is in family x. I came across this paper where they are speaking about an SR-like protein. They said:
While budding yeast lack alternative splicing, they do have three SR-like proteins: Npl3, Gbp2, and Hrb1. However, these have been best characterized for their roles in mRNA export, leaving their potential roles in splicing largely unexplored.
So in this case the protein has a serine/arginine binding domain and an RNA binding domain (just like SR proteins) but most of the biochemistry points to them being involved RNA export, not splicing (splicing, of course is what SR proteins are known for). So, they get the name SR-like. It will vary though. There is not a unified protein naming system