As you know I'm interested in grass, so have just done a small amount of reading about the different cuts of hay. It largely confirmed my understanding that, whilst there are differences in the different cuts, "what matters is the stage of maturity of grasses when cut and their proper harvesting to retain the ideal nutritional levels". At a basic level, it seems that all things being equal, a first cut will have longer stalks and will have seedheads, whilst a second cut will be shorter, with more leaves and will be softer. My take on that is that probably
bunnies might prefer the 2nd cut (and therefore eat more of it), but that 1st cut will be better for teeth and tummies due to more fibre.
https://www.bedlamfarm.com/2017/01/08/the-story-of-second-cut-first-cut-hay/
https://www.succeed-equine.com/succeed-blog/2016/10/07/myth-second-cutting-hay-best-horses/