Ken, you won't get any disagreement from me. On the old Monarch, I can't quite use the procedure I outlines above, but then, that's one reason I have the two lathes. Don't get me wrong, I can hit tenths on the Monarch too, mostly the same method, but there isn't the level of confidence there that is on the newer lathe, which by the way is an Acer. Pretty fair lathe, I think.
I try not to do the finicky work on the Monarch, unless the Acer has a setup I don't want to disturb. It's capable, and surprisingly repeatable for a lathe of its age. In reality, most of the work I do doesn't call for tenths tolerance performance. And the Monarch does just fine holding 0.002 for 'O' rings and the like. It's a pretty tight machine. One difference in the tow is probably the gear/bearing condition. The Acer will cut a finish that passes without much ir any polishing. I generally don't like papering on it anyway, but the Monarch just isn't as slick, so I do use paper. I prefer to use a flat file to back up the paper rather than just wrap it. In fact, this may be a good time to point out the danger of just throwing a strip of abrasive cloth around a spinning shaft. It will tend to be pulled in the direction of rotation, and given enough slack, can come all the way around and pinch itself up and snatch it right out of your hands, if your fortunate. Always be ready to let go of the paper if you insist on handling polishing like this. You can't win that wrestling match. I think most anyone who has spent time polishing shafting has had to give the paper to the lathe. I've been cut near to the bone by the edge of the abrasive cloth. Kind of painful, and makes you more cautious. And it should go without saying that wrapping paper around your finger to polish an ID is very, very risky. It can be very hard on the health of the knuckle joint of that finger. I keep a couple of "fingers" in the box for that when needed. One is simply a broomstick with a slot sawed into one end that I can wrap some cloth around and stick in the bore. Smaller than that, I have a couple of pieces of leftover Delrin set up the same way. For smaller yet ID's, where you can't get much pressure on the "stick", I use what most machinists around here call a butterfly. It's a 1/4" steel rod sawed to accept the cloth, and fit in an air motor. On this tool, the spindle doesn't even need to be spinning. Plus this can be used to deburr drilled/reamed/bored, etc. holes in nearly any part.