I am working on a motor very similar to the one you showed today. I have re-wired it 3 times, but I am a beginner and am still working on developing my skills. I was concerned since I was running out of lead into the wrappings whether I would be able to twist my wires together. I pre-twisted the wire I wanted to bring in so it had a spiral to it and then was able to get a decent connection to solder. This is a test motor to practice a few things prior to working on my Singer 15-125. So far, so good.
I took photos prior to taking it apart and when I match the same distance the belt pulley? was originally there is about 1/8 play in the shaft back and forth. The brushes are still in contact with the armature either way, but is that normal? I am not sure if there is a standard tiny amount or whether it all relates more to getting the positioning right for the belt on the machine.