How do I know which is the "correct" way to write the reaction?
How do I know which is the "correct" way to write the reaction?


There is no single correct way.  Picking a different way to write the reaction affects both $\Delta H_{R_j}$ and $Rr_j$ in opposite ways that cancel out.


  Does it mean that the reaction rate should be specific to R1 or R2?


Yes.  The unit of your reaction rate $Rr_j$ is $\frac{\textrm{kmol}}{\textrm{kg}_{cat}}$.  But the key question is kmol of what?  A good way to think about the answer is "kmol of things that have a stoichiometric coefficient of 1 in the reaction".

So if the reaction is R2, then the rate of the reaction will be the rate of disappearance of $\ce{O2}$.  The rate of formation of water will be twice the reaction rate $Rr_2$.  But if the reaction is R1, then the reaction rate is equal to the rate of formation of water (or the rate of depletion of hydrogen).

But you don't directly measure "reaction" rates, you can only measure the formation or disappearance of products or reactants.  So say you measured the formation rate of water.  If you decide the reaction is R2, then $Rr_2 = \frac{1}{2}R_{\ce{H2O}}$.  If you decide the reaction is R1, then $Rr_1 = R_{\ce{H2O}}$.  The factor of $\frac{1}{2}$ will cancel out the factor of two in $\Delta H_R$.  

So in the end it shouldn't matter what choice you make.

For the exact same idea in a totally different chemical context, see this se.chem question on electrochemistry.