Do this.

Get in car, turn key to START, if the starter does not engage, release key so it goes back to "on".
Wait 30 seconds.
Does the engine fan warp up to high speed? (You WILL hear it.)

If yes:
It is almost certain the RCM's engine control relay output is bad. THis can be due to fractured solder on the RCM, bad relay on the RCM, blown fuse number 2 on the RCM (not your issue because yours starts at times).
A replacement RCM would probably fix this. It is POSSIBLE you have a low battery, but in such cases, letting the car sit won't make it start later on. My money is on the RCM.

If no:
We have some more detective work to do...