edit:
on your side issue,
when you check the starter wire for voltage while (it is disconnected from the solenoid) in start mode it should be no less than battery volts. A bad connection or intermittant connection in the circuit maybe?

Also does it have a neutral start switch on the transmission?
sometimes that switch will be tied into the starter circuit and if it has alot of resistance it can cause this issue.
I had a toyota do me like this and it about drove me nuts till i found that bad switch.