From the thread linked above, I'm very happy with my second hand RX100 Mk 1.
It's compact: about 4" x 3" x 1"
It does long exposure: 30 seconds in the settings and a "bulb" I haven't figured out yet.
The autofocus is fast: I don't know how it's measured but it's quick.
The sensor is big: (ish) makes good use of the light it gets and minimises noise in low light. Probably some other stuff I don't understand.
On Ebay, anything between £170 and £240.
That's by no means a recommendation based on a thorough understanding of the camera market, but I have one, and I know I like it.
*better options may exist(*