Hi, To add. Have you checked voltage @ the battery when you hit the start button? Was the click from the relay or solinoid?
I've seen starters internals get fryed & drops the voltage like a stone. Just like a bad battery. But yours didn't as the headlights stayed lit.
If it stays up. Checking volts at the starter large lug. When you hit the button. Might be the starter solinoid itself, internal burnt contacs. Try using a jumper wire from the batt pos lug to the starter solinoid small terminal. Green wire on mine. That would energize the solinoid. If the solinoid was good, you would hear it click in. Starter would crank over. (That would bypass the switch & relay.) Test voltage @ the solinoids small wire to see if it matches batt volts when you press the start button. At least you would know then if the probs in the button, solinoid, or wiring.
If you want to take the solinoid apart to look at the internals. Leave the lock nut to the case alone. ( Where the pos cable attaches. ) Till you get the front 3 screws out & pull the guts out to inspect. You can loosten the pos cable & take it off, just not the case nut below it. Then loosten the nut close to the case. After the guts are out. Book said to do it in that order.
If you use an ohm meter. You can test ohms read between 2 contacs on the same wire. 0.0 is a good contac. But the ohms test is not as good as a drop test.
How to do a drop test->
http://www.aa1car.com/library/voltage_drop_testing.htm
Editing-> Relay clicks. Start switch to solinoid. Test for voltage to relay pin #30. Yes? Voltage present pin # 87 when the starter button is pushed? Voltage present at green wire @ solinoid? All yes, then hunt at the starter.
On the solinoid clicks. Says do as drop test batt pos to relay terminal on starter, then batt to starter terminal. The drop test should show less than 1v drop if good. Next is the ground. Drop test batt neg post to starter studs or bolts. Same deal on less than 1v.
If it is the solinoid, book says there's a repair kit for it. Dave