Boolean Expression to Ladder Translation for PLC Help

Hello! Please see the attached exercise attempting to show a technique for describing a finite state machine with two states (on and off) controlled by two switches (S1 and S2), using that graphical depiction to generate a boolean expression for the conditions resulting in those two states, and finally converting/translating that boolean expression to a ladder diagram.

At first glance it looks like I did it right, but something is missing in the final translation because in this example if we are already in the "OFF" state and attempt to transition to the "ON" state by turning input S1 "true" (see rung #4) the "OFF" input also on rung #4 will become "false" because of the NC S1 input on rung #1 before the "ON" output has a chance to latch via rung #4.

That's a lot of jargon but in short, I am hoping someone out there knows specifically how to translate the boolean expressions to ladder language. In the end I suppose it should look like the classic start/stop circuit you see on the first page of every ladder example. Thank you!

For what you're describing you don't need any kind of latch if I'm following correctly. "ON" is only true when S2 is off (NC) and S1 is on (NO), so your rung is just a NC contact for S2 and a NO contact for S1 to the ON coil. And then "OFF" is only true when S2 is on (NO) and S2 is off (NC), so your rung is a NO contact for S2 and a NC contact for S2 to the OFF coil. You haven't defined behavior for when both switches are either on or off, but neither case will have the ON or OFF coils energized.