Did you pull the armature out to see if there is a centrifugal switch?
Most that are capacitor run, the windings may be the same.
That original schematic shows the original typical set up for these lathes.
On the start relay, the retaining contact is physically made by the start button pushing the relay armature over by the button, a cheap way of doing it.
The reversing switch was a rotary drum style switch.
You will notice that the start winding is connected permanently to the Hot conductors from the start/stop sw. and only the run windings switched/reversed. for FWD/REV.
Al.