I swear I’ve heard both definitions for this word, but let’s see, shall we?
To me (UK), “starkers” is specifically and only ever ‘naked’.
“Stark raving bonkers” is ‘insane’, but I’ve never heard or seen it shortened.
This. I would have gone with “Stark raving mad” for the second example, but that’s even further from what the OP’s going for.
Aussie. Same answer though probably not bonkers, just stark raving mad or a stark raving lunatic.
To me, (USAian) it could be either naked or crazy, depending on context. Or both, as the two situations aren’t always mutually exclusive of each other.
If Jim is starkers, he’s definitely in the naked nudidity.
If he’s stark raving mad, he’s mental and hopefully fully clothed. Something with long sleeves that buckle at the back would be comforting fashion statement. For me, not him.
If you said he IS starkers, I would assume he’s insane. If you said he WAS starkers, I’d assume he was naked at some point in the past.
This!
It depends on context. Stark raving mad vs. stark naked. “He’s starkers!” is not heard much around here, usually he’s either nekkid as a jay bird or just plain crazy, but he could be both.
US, Deep South here.
I’ve thrown over ‘starkers’ for the more current ‘nakies.’
USA-type. Never heard the term before this thread.
It sounds Brit / Aussie / Kiwi / etc. to me but could mean darn near anything. My first guess was it was a new hipster terms for people with very black-and-white political / social opinions.
USAian here (good term, that, cochrane). 
starkers = naked
barking = nuts
Lots of BBCA watched. And Harry Potter movies.
starkers = stark naked rather than stark raving mad or stark staring mad
I’ve only ever heard “starkers” refer to someone totally naked.
Starkers have always meant naked to me, and I’m from the deep south of GA as well.
On an aside, am I the only one who misread the title of the thread as poll strokers?