Lagrangian: when to discount budget constraint?
The constraint is part of the objective function in each period. So, naturally it is subject to discounting, and so it must "be multiplied by beta".  

So the fist setup above is wrong (or just a typo: one should check whether in the specific publication the solution given follows the above image or in reality it implies that the constraint is multiplied by beta).

The "third case" the OP mentions in his post is a different matter. There we "compact" the constraint into one, so here they are outside the sum and the intertemporal discounting, with the understanding that the single multiplier is a present-value multiplier in this case. This is ok, it is done for example on certain macroeconomic models related to optimal taxation.