Interesting how the railways are embracing International when over the last few years airports have realised what a pointless word it is and dropped it (eg Bristol, Cardiff). Why not Fishguard International or Holyhead International? And it's only the British section of Eurostar services that names its stations International.
Is it perhaps logical to add "International" (or indeed "National") where there are multiple stations / terminals for within and without country. So Harwich International v Harwich Town, and St Pancras International v St Pancras Domestic - sensible. And now Fishguard Harbour perhaps should be renamed Fishguard International with the opening of a new station at Fishguard and Goodwick.
Cojoined countries in Europe tend to have a lot of traffic crossing borders and in places like Luxembourg, all stations would be "international" ... so little point and even in a big country like France, there's a multitude of International services from various Paris stations.
Airports? In the
UK▸ , most have International flights so it's taken as read, perhaps. You will find Washington National in the USA, and perhaps some others ...
Just musing .. no real answers.