I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has homogeneous coordinates $H:=\{x_i : i=1,\ldots, I\}$ equipped with a number $R$ of equivalence relations $$ (x_1,\dots,x_I)\sim (\lambda_r^{Q^{(r)}_1} x_1, \ldots, \lambda_r^{Q^{(r)}_I} x_I), \\ $$ for $r=1,\dots, R$ with the weights $Q^{(r)}_i\in \mathbb Z$ and $\lambda_r\in \mathbb{C}^* = \mathbb{C}-\{0\}$.
They go on to say (above equation (7)) that for each divisor $D_i$ of a toric variety, there is a line bundle $$\tag{7} L_i={\cal O}_X \bigl(Q^{(1)}_i, \ldots, Q^{(R)}_i \bigr)\; . $$
It is not clear to me why the weights $Q^{(r)}_i\in \mathbb Z$ govern the classification of line bundles, but this seems to imply that for a particular toric variety, $X$, the Picard group is $$ Pic(X)=\mathbb{Z}^R. $$ However, I have also read here -Reference for Weighted Projective Stacks - that for weighted projective spaces, the Picard group is cyclic. (Edit: It seems that this link talks about weighted projective stacks, which are not toric varieties.)
What is the rationale for describing line bundles in terms of weights as in equation (7), and how does one find the Picard group of a toric variety in general?
Edit: Fred Rohrer suggested in the comments that I look at Ewald's Combinatorial Complexity and Algebraic Geometry. On page 273 we find that for an arbitrary $n$-dimensional toric variety, $Pic(X)=\mathbb{Z}^{k-n-\lambda}$, $k$ being the number of 1-cones of the corresponding fan $\Sigma$, $\lambda$ being the total dimension of the spaces of linear dependencies of generators of all maximal cones which are not simplex cones. This, however, translates to $$Pic(X)=\mathbb{Z}^{R-\lambda},$$ which is different from the expression above. Why is there this discrepancy?