What is the difference between a phosphotransferase, a phosphatase, a phosphorylase and a kinase?
Don't get caught in the linguistic trap of attaching too much significance to the precise meaning of any one piece of scientific jargon. Nomenclature, the system of naming things, utterly sucks in biology, and it sucks even harder for protein biology. There is no one universally accepted classification and naming system for proteins, although there have been plenty of attempts.

For example, phosphotransferases are classified as EC number 2.7. However, Glycogen Phosphorylase is classed under 2.4, which might make you think "well, clearly kinases are enzymes that transfer phosphate groups from organic substrates such as ATP and are classed as 2.7, whereas phosphorylases are enzymes that transfer from inorganic substrates and are therefore classed as 2.4". But wait! RNase PH is a phosphorylase that's classed under 2.7 and uses inorganic phosphate, so um... yeah.

In any case, don't get too hung up on the nomenclature. Unless you're working in an enzymology lab just say "kinase" and everyone will know what you're talking about.