Thanks Leon - the more I look about here the more I spot too.
The back end of mine is all D-tube, as per the 1913 Brown Brothers page (which shows a frame with a pedal bottom bracket), and also has the front chainstay lug shown in the bottom image you posted - which, again, appears in the image of 1912 Sun lugs on page 107 of Rob Saward's green book, and is appropriated by a New Comet leaflet with 'Haden's Motor Sets' added to the bottom of image (it's described as being for machines without pedals).
Another page shows the cross-over linkage - there are variations of this though - Suns and a Precision had the crossover about halfway down the rear engine plate, and a 'below centre' brake shoe. There's a Regal outfit with the high-level crossover and 'above centre' brake shoe, and one of the Regal Green images from the time has the same high-level cross-over.
An image of a ladies' model Regal from showtime 1913 indicates that it has the same rear stays as mine - the 'eye' for the starting handle, and no provision to attach the carrier to the seat-stays (most had a 1/4" stud that went through the cross-piece where the mudguard bracket fits - my frame has a thin cast 'web' and no attachment points).
Why Armstrong? They all had a flatted spindle, 7/16" across flats: the larger Sturmeys (JA and JS) all had a round spindle, which I think was 9/16" diameter, and just wouldn't go in the drop-outs I have.
Plus, I've dropped a restored one in, and it goes straight in, with enough room for a torque arm on the drive side (as shown on the 1913-for-1914 Hampton which appeared in the press of the time).
Looking at the lists of machines for 1913 and '14, the Sturmey hub was the most common offering with the Precision engine (only a small percentage of machines offered in the UK had the Precision engine).
But, from small ads, it appears that 1913 Regals were offered with the Armstrong gear.
Haven't found a number yet: even if I did, I'm not sure there would be any records to check it against.
Nick