Say your HMS Frigate rolls up on an American merchant vessel and her boarding party demands to "reclaim" all British-born able-bodied seamen. How would the birthplace/citizenship of individual members of the American crew be identified?
From reading Ronald Utt's Ships of Oak, Guns of Iron , it seems to me that the British, who insisted that their impressions were limited to those they believed owed loyalty to the Crown, must have claimed a use of some sort of system to distinguish the sailors in question from other sailors.
Foreigners were technically protected from impressment, but this legality was often ignored and those who couldn't prove their citizenship would be pressed, as often happened to American sailors. Britain viewed all Englishmen aboard any ship, even foreign ones, as eligible for impressment.
There were also individuals protected from impressment: apprentices indentured to a master, seamen with less than two years' experience at sea, fishermen and other trades connected to maritime trade and industry (sailmakers, shipwrights), which were considered economically vital and and not to be pressed. But this didn't prevent press gangs from scooping them up, as such individuals needed to present documentation immediately upon demand by the gangs, proving they were in a protected class. If they failed to do so, they would be pressed.
As many as 15,000 American sailors were pressed into British service during the French Revolutionary and Napoleonic Wars (1793-1815). Due to the high turnover of men due to loss in battle, low pay or desertion (which regularly numbered in tens of thousands) the Navy was always short of hands, so the press gangs were always working. They were desperate for sailors throughout this period in order to keep their navy afloat and in action. It was seen as their right by custom, which had existed as far back as the medieval era.
The impressment issue was cited as one of America's grievances for the War of 1812, but Britain -- in a titanic struggle against Napoleonic France (and having been at war with France for nearly a quarter-century) -- saw the issue as an opportunistic pretext for the US to go to war. They were outraged that America would attack them, when Britain was in a fight for survival.
In the Treaty of Ghent in 1814, it should be noted that the peace treaty actually mentioned none of America's main grievances for war: neutrality of its vessels and impressment. Britain eventually would stop impressment of sailors, on its own accord, though the practice did continue even after Ghent.