What does `weakly gravitating' mean? When relativists like Bousso (see for instance https://arxiv.org/abs/hep-th/0203101) talk about `weakly gravitating systems', what sense of weak gravity is usually meant?
(1) Post-Newtonian approximation
(2) Linearised gravity
(3) Something else?
 A: You are confusing techniques for making a calculation with a description of a physical system.  Weakly gravitating means what it sounds like: The gravitational forces are "small" deviations to a flat (Minkowski) geometry.
On the other hand, "post-Newtonian" methods and "linearized gravity" are mathematical models / computation techniques that may (or may not) be used to compute a quantity of interest, usually when gravity is "weak".  You could just as well use the full Einstein equations for your calculation though, even though the system is "weak" and not all terms will contribute equally.  Regardless of what you do to try to compute, however, you don't change the physical system's characteristics. (You may get better insights into those characteristics by choosing a good method of calculation, of course.)
A: Technically a weak gravitational field means a spacetime geometry where the metric tensor $g_{\mu \nu}$ can be written as
$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$
where:
$\eta_{\mu \nu} = diag(-1, 1, 1, 1)$ Minkowski metric tensor (flat spacetime)
$\vert h_{\mu \nu} \vert \lt \lt 1$
That is the correction to the flat geometry is a small perturbation.  
This allows for a simplification in the calculations, as you may neglect terms quadratic in $h$.
