The 8.8 (.3465") would be the bore dia which was the way the proof house marked bbls during this period. 9.3 (.366") is groove or bullet dia, though I believe most 9.3x72R rounds actually used a .364" bullet. Is there a mark denoting the bullet used, my 9.3x72R bbl from a 1913 drilling is marked KMG/12.7gr IE copper jacketed bullet of 12.7 grams wt (196 grains). Lookung forward to the better pics, but appears a very nice gun.

Miller/TN
I Didn't Say Everything I Said, Yogi Berra