Why are there four R2s? Which one should I use?
The problem of the standard r2 under many fixed effects is that its value will mostly be driven by the fixed effects and not by the regressors of interest. As a solution, the within-r2 computes the r2 of the regression where every variable has already been demeaned with respect to all the fixed effects.
Therefore, on top of the usual r2 (e(r2)) and the adjusted r2 (e(r2_a)), we have the within-r2 (e(r2_within)) and the adjusted within-r2 (e(r2_a_within)).
If what you want an acid r2 that relates only to the variables of interest and is not driven by the fixed effets, I would suggest the adjusted within-r2.