On-Line Diameter Measurement

Optical. Definitely, I'd go for optical. Something like a sensor properly mounted. There are several models available. I'd start to looking at globalspec for optical micrometers.

If I understand your text the tolerance field is ±0.1 mm for a 512.25mm nominal diameter.

Optical can be sensitive, on the lathe, to dirt and give wrong values. I would prefer either a LVDT or magnetic pickup or even a pneumatic one. The equipement should be set for the nominal diameter and measure with high resoulition only the deviations. How the system has to be put on depends on the kind of lathe you have and the possibility it offers.