About that web page which describes various jackets, I found it curious that they said that the Regulation Doublet was so-called because it was the regulation doublet of the Highland regiments. As far as I know, no jacket resembling that photo has ever been worn by any regiment in the British Army. The jacket in that photo seems to have the lapels of a Prince Charlie but the tashes of a Highland doublet.
It's interesting to compare modern offerings with the sorts of Evening jackets offered in Highland Dress catalogues from the 1920s and 1930s.
In this catalogue image, dated 1926, you can see the only two Evening styles offered by RG Lawrie at that time, the "coatee" and the "doublet".
My Wm Anderson & Son catalogue from 1936 has these two styles as well, also called simply "coatee" and "doublet", but also has a third, of which they say:
"The Doublet is less popular than it once was, but is suitable particularly for older men. To get over the objection of the rather heavy skirts we have designed a modified doublet known as the Kenmore Doublet. This makes a neat and shapely coat, and has proved immensely popular since its introduction...
It is a simplified form of the old-fashioned Doublet and may be made with a stand collar or with an open collar."
(This is the only Evening jacket appearing in my old catalogues which requires a belt.)
My Paisleys catalogue from 1939 gives these three styles plus the "Montrose jacket", standing collar, double-breasted, and lacking skirts.
Of these styles I've only come across the "Doublet" in 19th century photos, making me think that the Kenmore, Montrose, and Coatee (Prince Charlie) were devised in the 20th century.
Note that in the 19th century the "Doublet" had an ordinary open collar such as appeared on suit coats of the time, not satin long lapels like the Prince Charlie and modern Regulation Doublets.