solenoid, starter or both? Best guesses welcome

Hey. I don't have a multimeter yet (my tool kit is growing as fast as I can). I'm looking for a best guess so far:

cleaned the contacts in the starter button (rust gently removed, etc) but the small brass rectangular plate right behind where the starter button doesn't go anywhere. Is that what it's supposed to do/not do? When I start from cold (charged battery), I get clickity clack clack "I wanna start, I don't wanna start" then I release the button, push it in again and get absolutely nothing, then I get a teasing turnover, then nothing, then it turns over (with about 10 seconds hanging on the starter-also recently rebuilt-and a quick choke) and we're fine.
Since I have to order a new choke cable from Chacal, I'm trying not to overdo it as I tend to flood it. the throttle cable moves freely and releases (don't think the spring is lazy, but I still get high revs...hopefully just need a carb tune up). OY So much to do..what first?
Does this sound more like a solenoid problem or a starter button problem..or a different electrical problem (guy before me replaced the voltage regulator which was trashed from starting form a half-charged battery, said he????
tks. for helping me prioritize!
;-)

 

I would take off all the leads to / from solenoid / starter & battery, clean with emery paper & re-fix, make sure all the connections are good & tight.

 

So far, so good; fingers crossed till the end of the season!