I myself am rather unfamiliar with oil as a lubricant. I have the impression that the vast majority of stones today are lubricated with water. And for woodworkers, the proximity of oil is not exactly enticing. I also have a Tormek grinder, so I'm set on water from the start.
One could almost have the impression that mainly only natural stones still require oil (e.g. Arkansas stones). But that's not quite true. E.g. Norton even offers stones that work with oil and water. Norton even describes that their stones are pre-soaked with oil by default. However, they also warn that they work worse with water.
Oilstones are generally considered harder and rarely require dressing. I must admit, however, that the rust sensitivity of diamond plates would possibly be an argument to rely on oil throughout. Unfortunately, I do not know exactly what effect the type of oil has. Apparently, a thick oil makes the stone "finer".