[IanH]

So the light don't dim or go out when you turn the key ( and still13 v)
Either you have a very bad contact, clean battery posts, check cable at solenoid
Or the solenoid and or starter are not drawing power, ie they quit.

 

[IanH]

You have to jump the solenoid to pull in the starter pinion, then main contacts close.
Try 12V to the normal Small terminal.
If it works then its your ignition switch
Also try this.
Have someone hold the key over in the start position and hit the main starter body with a hammer, give it a good whack.
If it starts replace the starter.

 

[IanH]

Unfortunately you haven't found the cause unless it was just a bad connection.
If it is the starter, it will be back.................

 

[IanH]

I would say that problems are in order of frequency reported on the forum
1) starter
2) ignition switch.

Although you don't seems to want to believe this is the starter, they do this, work for a bit, then not start.