Yes the grease nipple is to regularly grease the diff. my hope is the excess will also work its way out between the two shafts on the left hand side lubricating the bronze bush out there. To assist that I replaced the o-ring that seals the sprocket to the drive shaft but not the internal one that's suppose to seal the outer shaft on the left to the axle shaft.
Yeah the grease is lithium, its a better quality one fuchs epx2 and I have no idea what is recommended for the diff. When I pulled it apart I expected to find a molybdenum grease but the grease just looked generic so I lithium top up more regularly should be fine.
I'd do something similar to the steering for sure if it gives me trouble. This ol mower has saved me some coin over the years, even these repairs ... in parts have cost me relatively little verses having a hydrostatic bail out.
