In Royall Tyler's 1787 comedy of manners, Jonathan, a likable country bumpkin type in New York for the first time, hopes for "a drab of cyder with a pepper-pod in it." I assume he's talking about mulled apple cider, and the OED suggests that the pepper would be a hot chile. What sort of hot chile would Jonathan be growing or eating at this point in history?
That is going to be very difficult to establish with any degree of detail.
According to John Smith (1882), pepper were very broadly classified as either "pod pepper" or "cayenne pepper"; the former referred to any variety of fresh pepper while the latter was used to dried and ground varieties. Both sweet and hot varieties were available and adopted by Europeans.
Otherwise, only one of the 5 species of pepper domesticated by american natives was widely grown abroad and gave rise to the majority of the varieties grown in the Western world, and that is the highly variable Capsicum annuum Debra Fett (2003). As to varieties grown in the US, they seem to have been introduced from Europe instead of directly from Meso America (no matter how counterintuitive that may seem), which might help narrow down the possibilities a bit (see Heiser and Smith (1953)).
TLDR: Although it is not possible to nail down a specific variety, that pepper would have probably have been variety of C. annuum of European derivation.