I think a bit of both. The Original 50's or 60's I played at CME the other day, neck was like a baseball bat. I could barely play it. I think I might go today and try the American Pro or what not.
I had a Classic 50s and a '57 AVRI; I found my hand definitely adapted to the wider nut, but ultimately the American Standard P neck is where I live, so both of those were replaced eventually by an American Standard. The American Professional plays like a dream, for my hand; it almost feels like it plays itself.
Because of the narrower nut I find the Jazz to feel much more like a baseball bat, and similarly I found the 50's spec width to feel flatter, even though they're both about the same depth.
Slim to me means depth, front to back - I find the Geddy neck is like this, it is slim and made me smile when I first picked one up. I'm not sure how I'd take to it during a four hour gig though. I've owned one of the Squier Vintage Modified fretless P's, and found it's neck to be slim/flatter.
Frets make a surprising difference too - the Aerodyne I had played really nicely with vintage size frets and a J width neck, very comfy, but the same frets on the AVRI and the Classic 50s weren't doing it for me, and the tall narrow frets on the Pro feel amazing.
So I guess my point is you have to play the field, but hope this helps in narrowing down the field somewhat.