First cut can tend to be a stalkier coarser hay, high in indigestible fiber and low in nutrients and protein. Second cut is usually a mix of leaf and stem with a good mix of indigestible fiber and nutrients, with a moderate amount of protein. Third cut is usually a softer leafier and richer hay, higher in protein and lower in indigestible fiber. But even then this is just a generality, as the final quality of the cut will depend on what the growth stage is when the farmer finally harvests the hay.
First cut if cut when it is mostly matured and coarse, because of it's coarseness some rabbits may not like it, though the high amount of indigestible fiber can be good for increasing gut motility when fed temporarily, but the low amount of nutrients and protein could cause weight loss and nutrient issues for rabbits on a minimal pellet diet when this hay is fed on a continued basis. In some instances it may also lead to an increased risk of cecum impaction or fatty liver disease, due to the low nutrients affecting weight loss and movement of cecotropes through the cecum.
Second cut when harvested at the appropriate time where it is not fully matured but still has some soft leaf with the stems, is mine and my rabbits preferred cut. It has a good mix of coarse stems for the needed indigestible fiber to improve gut motility, and leafy strands to provide the nutrients and protein good for weight maintenance and health.
Third cut will usually be quite rich and leafy. Though this is sometimes a much preferred cut by rabbits, it can also cause some issues like upset stomach and/or excess cecotrope issues, because it is very rich in nutrients due to it's early growth stage when it's cut. It won't have a lot of indigestible fiber for good gut motility because it is primarily soft strands with very little to no coarse stems.
Which will be the best cut all depends on the growth stage of the hay when it was cut and also what your rabbit is tolerant of or prefers to eat. Whatever the cut is, I want it to be a medium mix of leaf strands and coarse stems, with no brown hay mixed in.