Comparison to other work
BCMS20 Appendix A.2
Appendix A.2 of BCMS20 describes a polynomial commitment scheme that is similar to the one described in BGH19 (BCMS20 being a generalization of the original Halo paper). Halo 2 builds on both of these works, and thus itself uses a polynomial commitment scheme that is very similar to the one in BCMS20.
The following table provides a mapping between the variable names in BCMS20, and the equivalent objects in Halo 2 (which builds on the nomenclature from the Halo paper):
BCMS20  Halo 2 

$S$  $H$ 
$H$  $U$ 
$C$  msm or $P$ 
$α$  $ι$ 
$ξ_{0}$  $z$ 
$ξ_{i}$  challenge_i 
$H_{′}$  $[z]U$ 
$pˉ $  s_poly 
$ωˉ$  s_poly_blind 
$Cˉ$  s_poly_commitment 
$h(X)$  $g(X)$ 
$U$  $G$ 
$ω_{′}$  blind / $ξ$ 
$c$  $a$ 
$c$  $a=a_{0}$ 
$v_{′}$  $ab$ 
Halo 2's polynomial commitment scheme differs from Appendix A.2 of BCMS20 in two ways:

Step 8 of the $Open$ algorithm computes a "nonhiding" commitment $C_{′}$ prior to the inner product argument, which opens to the same value as $C$ but is a commitment to a randomlydrawn polynomial. The remainder of the protocol involves no blinding. By contrast, in Halo 2 we blind every single commitment that we make (even for instance and fixed polynomials, though using a blinding factor of 1 for the fixed polynomials); this makes the protocol simpler to reason about. As a consequence of this, the verifier needs to handle the cumulative blinding factor at the end of the protocol, and so there is no need to derive an equivalent to $C_{′}$ at the start of the protocol.
 $C_{′}$ is also an input to the random oracle for $ξ_{0}$; in Halo 2 we utilize a transcript that has already committed to the equivalent components of $C_{′}$ prior to sampling $z$.

The $PC_{DL}.SuccinctCheck$ subroutine (Figure 2 of BCMS20) computes the initial group element $C_{0}$ by adding $[v]H_{′}=[vξ_{0}]H$, which requires two scalar multiplications. Instead, we subtract $[v]G_{0}$ from the original commitment $P$, so that we're effectively opening the polynomial at the point to the value zero. The computation $[v]G_{0}$ is more efficient in the context of recursion because $G_{0}$ is a fixed base (so we can use lookup tables).