I agree with all the above sentiments. As a beginner, on a lathe of unknown quality, with unknown wear, you will struggle to improve on the one thou runout and quite possibly make it worse. Plus there is the flatness issue, even more difficult to achieve than the runout due to wear on the carriage way surfaces and cross slide.
Certainly try the suggested take the faceplate off and clean the threads and register faces etc and retest again, several times. You might be surprised at the variation.
The layer of rust on the faceplate (or is that exaggerated in the pic?) could make that much difference alone.
Headstock bearing adjustment and wear could make that much difference too.
If you just want to have some fun, bolt a disc of metal to the faceplate and take some test cuts across it and see how it turns out. When you can get that perfect, unscrew the faceplate and clean the threads and put it back on and see if its still perfect. When it repeatedly is, you can think about facing the face plate.
To measure flatness, you need to either use a straight edge and feelers across the face. Or you mount a dial indicator on the cross slide and run it across the BACK half of the machined face, ie between the centre line and the REAR of the machined face. If you just run on the front half, the dial indictor just follows the same path as the tool and reads 0-0 regardless of any angle on the face. But the back half is angled the opposite way. So the true variation is half the reading over that surface. Flatness across the faceplate should be either perfectly flat ie 0-0 reading or very slightly concave, probably within a thou or so for home hobby work on an old machine. This is because a job will sit firmly on a concave surface but will rock on a convex one.
Personally, I would leave it as is and carry on. If you want to machine the ends of your large round bar any closer to parallel than one thou, use the suggested method of between centres and face both ends in one set up.
Then you need to work on your ability to measure less than one thou with a large micrometer. (Digital calipers really aren't accurate enough.)
Edited By Hopper on 15/08/2020 02:22:18