Having studied supersymmetry in $d=4$, my understanding is that we count supersymmetries by the number of pair of complex supercharges $$ Q_\alpha^I = \begin{pmatrix} Q_1^I \\ Q_2^I \end{pmatrix}~,~ \bar{Q}_{\dot{\alpha}}^I = \begin{pmatrix} \bar{Q}_{\dot{1}}^I \\ \bar{Q}_{\dot{2}}^I \end{pmatrix} $$ which are Weyl spinors and hence each has 2 complex off-shell degree of freedoms (dof) and 1 complex on-shell dof. Minimal supersymmetric models have $I=1$, and are called as $\mathcal{N}=1$ susy models. My questions are as follows
When we say that $\mathcal{N}=1$ susy models in $d=4$ has $4\mathcal{N}$ real supercharges (see e.g. here), is it because $Q_\alpha$ has $4$ off-shell real dof, or because $Q_\alpha$ and $\bar{Q}_{\dot{\alpha}}$ have a combined total of $4$ on-shell real dof ? I think the second one makes more sense because supercharges are Noether charges which are conserved only on-shell.
On the other hand, in $d=1+1$, susy models containing $4$ complex supercharges $Q_+$, $Q_-$, $\bar{Q}_+$, $\bar{Q}_-$ are said to have $\mathcal{N}=(2,2)$ supersymmetry (wiki) , i.e. for this case one counts each single supercharge instead of pairs in contrast to $d=4$ case. So, why is this difference in the conventions ?
