I think it gets down to what you're used to and the feeling that goes with it. I played with a Black Dot shaft at around 12.3 mm for some years. Then I ordered 2 shafts, The Pro at 12.5mm and the Ultimate Weapon at 11.65mm. When I first tried the 2 new shafts I had some troubles with the 11.65mm UW. I liked the taper/diameter close to tip of that one better, though.
It took me much longer to get used to the 11.65mm. Now I only play with that one.
I believe it may be due to the optics aiming with the shaft. As everyone else has said, you see more of the cb with the thin shaft. However, I may also have gone further outside with the thin shaft. The optical illusion may be that a 13mm shaft is "extending" like 6.5mm to the outside of the actual spot you hit. The 11.65 extends only 5.8mm and thus it seems that you are not as close to the miscue limit. Obviously, this is not scientifically correct if you hit the exact same spot on the cb but I believe I may have gone further outside with the 11.65 and thus had more spin - but more miscues as well. If that makes any sense.