Leo,
What you write regarding import/export is good advice.
If I may slightly correct & expand on your first line though: There aren’t two types of UK shotgun certificate – the single type enables the holder to both carry & acquire shotguns.
Also a visitor to something like a shooting school, clay pigeon club or organised game shoot (for instance) can shoot a shotgun without themselves holding a certificate if accompanied by the landowner, their agent or an instructor (who must hold a certificate & provide the shotgun).
Visitors to the UK who wish to arrange their own unaccompanied shooting can apply for a ‘Visitors Shotgun Permit’ (here’s a quote from a police website):
“The holder of a visitor's shotgun permit may have in his possession, purchase or acquire shotguns”.
So in a nutshell – it’s quite possible for a visitor from the USA to get a permit in advance; come to the UK & buy a shotgun; use it for a while & then (preferably through a dealer) ship it back to the US (a wise buyer would purchase from a dealer who could also arrange the export as part of the deal).
Regards
Russell