While I'm by no means as experienced as other posters, this is the way I do it: set toolrest as close as possible use a pencil to mark the center of the spinning piece. If I'm off, I erase it and try again until I have a neat point set the calipers to the radius of the piece with the lathe running and hte calipers flat on the rest, dig one point of the calipers into the center, just far enough for it to hold still then slowly push the other point into the work.
If you're just making an approximate mark, you can skip the pencil stage, the first dig into the wood as close to the center as possible by guesswork will hold the point in place.
Incidentally, any way you do it, this practice vibrates the calipers. If you're using calipers with one of those tiny set screws, make sure you tighten the screw hard every time, or you'll be looking for it among the shavings on the floor.
Only one P in my real address/ Un seul P dans ma véritable adresse