It's a bit complicated, because the increase in stress on the fault from the weight of the reservoir can actually be primarily an increase in normal stress, i.e. an increase in the clamping force preventing the fault from slipping (i.e. the crust on either side of the fault sliding, i.e. the earthquake), so the weight of the reservoir can actually impede, rather than promote, seismicity. However if the geometry of the system is just right such that the increase in shear stress from the weight of the reservoir can increase on the fault, then the reservoir could really trigger the earthquake. Typically, though, the increase in pore fluid pressure is the most likely trigger as it's not geometry dependent--pumping a fracture with fluids will always promote failure. This is what happens with the seismicity associated with natural gas fracking and subsequent wastewater fluid injection.
The thing that is hard to grapple with all of this is that changes in stress and fluid pressure on the fault are really, really small compared with the ambient stress and fluid pressure. These are tiny changes, but there is good reason to believe that many faults, especially those in seismically active areas, are very close to frictional failure (i.e., slipping and producing an earthquake) all the time. So therefore little things can perturb the system. However, it's worth keeping in mind that most of these small changes slightly increase or decrease the time to failure (as the primary loading that is causing the stress on the fault is probably continually happening at very low rates), rather than being the ultimate cause of the earthquake.
[1]: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C48&q=dam...
[2]: https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/200...