I believe that the Birdwood museum exhibit is a 500cc 3.1/2 hp. model i.e. the predecessor of the more common 600cc 4hp.
The first picture shown by Trevorp is also not a 1910 model but a later 2.3/4hp. The 1910 model is very similar to the 1911 model shown except that the rear fork on the 1910 extends back from the bottom frame tube instead of between the two frame tubes. There are other minor differences such as lack of a cast alloy cover over the timing gear on the flywheel side of the engine, sharp edges on the fuel tank, filler caps on the other side etc.
Interestingly the 1911 model shown is fitted with the very rare two speed gearbox. This was available as an after purchase bolt on "extra" and differs from the usual 2.3/4hp two speed gearbox in that the input and output are on the same side of the gearbox.
This gets us no closer to knowing what a model 'R' looks like. I also cannot find an illustration of this model, only a listing that says that a model R was made in 1913.