In griffith's intro to electrodynamics (4rth edition), ch. 3, pg. 121.

here is the second uniqueness thrm for the solutions to laplace's equation:

the only part i'm confused about is, in the beginning where he says "in a volume V surrounded by conductors and containing a specified charge density ρ"

Q: is the charge density ρ referring to that belonging to the region of space whose potential we're trying to determine? or is it referring to that of the conductor/insulator?

but then this leads to another confusion. I thought both uniqueness theorems were meant for the solutions to the laplace equation. The laplace equation however does not deal with charges inside the region.

so then, i guess, uniqueness thrm 1 is for both the laplace & the poisson equations while uniqueness thrm 2 is for just the poisson eqn?