TDI has some very good definitions, presumably applicable in the UK as well as elsewhere in the world.
Diving with redundant air source for recreational divers - SDI | TDI | ERDI | PFI
Pony cylinders (3-40 cuft) are categorized as
redundant air sources...not to be included in your gas plan. They are for emergency use. Yhey can be slung or back-mounted. How they are carried does not change their name or their use.
PADI also defines a pony as a reserve cylinder (6-49 cuft, 1-4 lt). The reserve cylinder is defined by its intended use, not its mounting.
NAUI also defines a pony as a "completely separate source of air," used "as a back-up system"
A stage bottle is part of your gas plan, and extends your dive...no matter how it is carried\, or whether you drop it or not.
See
Pony bottles VS stage bottles for a succinct description.
The point: ponys and stages are distinguished by their intended usage, not their method of being carried. In my travels around the world, this seems to be agreed. Here on SB, the argument comes up sporadically, often from UK divers who maybe don't get out and about much.