Angular momentum of a system about the center of mass

Note that $\boldsymbol p'_i:=\boldsymbol p_i-m_i\dot{\boldsymbol R}$. Therefore, $$ \sum_i \boldsymbol{r}_i^\prime \times \boldsymbol{p}_i-\sum_i \boldsymbol{r}_i^\prime \times \boldsymbol{p}_i^\prime\equiv\sum_im_i\boldsymbol r'_i\times\dot{\boldsymbol R} $$ which is easily seen to vanish, inasmuch as $$ \sum_im_i\boldsymbol r'_i\equiv0 $$ by definition.

Thus, the expression in the OP and the one in Goldstein are identical.