The pathing ( each of the cuts that the lathe will do ) is based on the profile of the part to be machined. The profile you draw is only the lower or upper half of the profile ( in many programs ). Thus both sides, when looking in plan or elevation view of the profile will have a radius. This is true, since as you said, the spindle is turning.
You must define to the program how you want to machine the profile. The machinng is usualy done by making more than
one machining sequence. ie; a rough cut to remove as much material as posssible in the shortest time, then a rough / finish pass to clean up the rough, then a finish pass ...which in some cases will only remove a small amount of material and may make a single cut following the contour of the profile. So you will need to change tools in order to accomplish the machining.
There are Wizards in Turn to accomplish simple machining tasks like your pictures and would recommend you fool around with them.
Hope this helps,
RICH