Not sure if the question was answered, but the OP mentioned having recenlt changed the starter. I recently had my starter rebuilt, and went through a similar situation to where my conclusion was "i had a bad rebuilt starter". After a while I figured out the starter was installed improperly (backwards).
Also, have a look at the black distribution box under the hood (front right location), and check your fuses. Swap your wiper fuse for the starter fuse to test if that's the cause. Examine the connection of the wires to the relay for any problems.
If you can confirm your ignition switch is fine, more than likely there is a relay problem.
To the LP - Sounds like it's still the ignition coil that's the problem. If it's installed properly, change your spark plugs (if needed) to the cheaper gold ones, and test your battery and alternator. When did you last change the spark plug wires? How is your temperatures..is it running hot?