Integral membranes proteins serve as transporters. Peripheral proteins serve as cell adhesion molecules, antigens and enzymes. So what about receptors? Which proteins carry out the duty of receptors?
All the receptors I know about are integral, transmembrane proteins. It would certainly be possible for a receptor domain to exist on a peripheral protein that interacted with a transmembrane protein, but I don't know of any examples of this: an extracellular peripheral protein would tend to float away, which is a bit of a problem if they aren't intended for secretion.
There are certainly peripheral proteins that associate with the internal parts of transmembrane receptor proteins and may influence their function, but they are not themselves receptors.
I'm not sure about your question about "which proteins" - those proteins are most often named after their ligand followed by the word "receptor", or based on their function that is triggered by binding their ligand, or both (i.e., the AMPA receptor is a glutamate receptor that is also activated by AMPA; receptor tyrosine kinases are a family of receptors that are tyrosine kinases; the individual members of the family are typically known by their ligand).