I've never ever heard "cote" used for anything but pigeons and doves. I think this is just sales-pitch labelling.
The Forsham item from the link is just *preposterous* IMHO, for anything other than conversation-piece housing for maybe 2-4 hens in an always-mild climate. Very silly. North Americans should note that Britain lacks raccoons or possums, and while in principle it is probably possible to build a coop on posts with enough overhang to defeat those climbers, the relatively small amount of overhang in the Forsham one would be pretty risky IMHO in North America.
If a person wants a dovecote-shaped chicken coop, just for purty, then my suggestion is to look at pictures/plans for actual garden-type *dovecotes* of sufficient picturesqueness and just rejigger the inside and opening sizes for chickens.
Pat