Interpretation of Coloumb operator in Hartree Fock equation
From the labels ($i,j$), I am guessing that $\chi$ is a molecular orbital (MO). When formulating the equations for molecular orbitals, one assumes that there are as many (spin-)orbitals as there are electrons. In any event, when writing down the Coulomb operator in the MO representation for the ground state, only occupied orbitals appear.
Therefore, I believe that the equations you are showing here are for the general problem, not just $\ce{H2}$. In some sense, the equations are the same for either, it is just a question of how many summands there are. Finally, note that the last equation you showed is valid for any system, but that it is actually part of a set of such equations, one for each MO.