If Im understanding what your asking it sounds like one of two possibilities. Stuck or broken starter button in the right switch assembly or failed starter relay.
a failed starter relay.
Follow the large red positive wire from the battery to a small square "box", that is the starter relay with the key off, test for continuity between the two large posts. If its open, your good, if you have any resistance value at all, the relay is bad

If not the relay... then disconnect the small two wire from the starter relay... that is the trigger wire for the relay.. now turn key one, motor no longer turn over right away without pushing the starter button? Good, now you know the issue is the starter button or the wiring along the way from it to the starter relay. Go up to the right hand switch assembly, follow the cable till you find the connector..and disconnect, following the wiring diagram, test the two starter switch wires for continuity, should be none (open) normally, and good continuity (closed) when you push the button.