Bishop

The "Bishop" appears to be a specific drink recipe and not a "category" of drink (like the Cocktail, Cobbler, or Sling). During the 1800's you could order a gin cocktail, whiskey cocktail, brandy cocktail etc. and because the term 'cocktail' referred to a category or style of drink, the bartender would apply that template and mix your drink for you. The Bishop however seems to have been made specifically, with port wine, and if you changed the base to something else, it would become a different drink. Lawn Sleeves was made with madeira, Cardinal with claret, and Pope with champagne.

One of the oldest, if not "the" oldest recipe for the Bishop comes from the 1827 booklet "Oxford Nightcaps", here is its entry for the Bishop in its entirety:

OXFORD NIGHT CAPS, &c. ~ BISHOP, OR SPICED WINE. Three cups of this a prudent man may take, The first of these for constitution's sake, The second to the girl he loves the best, The third and last to lull him to his rest. Ancient Fragment. BISHOP seems to be one of the oldest winter beverages known, and to this day is perferred to every other, not only by the youthful votary of Bacchus at his evening's revelry, but also by the grave Don by way of a night cap; and probably derives its name from the circumstance of ancient dignitaries of the Church, when they honoured the University with a visit, being regaled with spiced wine. It appears from a work published some years since, and entitled Oxoniana, or Anecdotes of the University of Oxford, that in the Rolls or accounts of some Colleges of ancient foundation, a sum of money is frequently met with charged "pro speciebus," that is, for spices used in their entertainments; for in those days as well as the present, spiced wine was a very fashionable beverage. In the Computus of Maxtoke Priory anno 1447, is the following curious entry; "Item pro vino cretico cum speciebus et confectis datis diversis generosis in die Sancti Dionysii quando Le fole domini Montefordes erat hic, et faceret jocositates suas in camera Orioli." "Vinum creticum" is supposed to be raisin wine, or wine made of dried grapes; and the meaning of the whole appears to be this : Paid for raisin wine with comfits and spices, when Sir S. Montford's fool was here, and exhibited his merriments in the Oriel chamber. Recipe    Make several incisions in the rind of a lemon, stick cloves in the incisions, and roast the lemon by a slow fire. Put small but equal quantities of cinnamon, cloves, and mace, and all-spice, and a race of ginger, into a saucepan, with half a pint of water; let it boil until it is reduced one half. Boil one bottle of port wine; burn a portion of the spirit out of it, by applying a lighted paper to the saucepan. Put the roasted lemons and spice into the wine; stir it up well, and let it stand near the fire ten minutes. Rub a few knobs of sugar on the rind of the lemon, put the sugar into a bowl or jug, with the juice of half a lemon, (not roasted,) pour the wine upon it, grate some nutmeg into it, sweeten it to your taste, and serve it up with the lemon and spice floating in it.    Oranges, although not used in Bishop at Oxford, as will appear by the following lines, written by Swift, sometimes introduced into that beverage. Fine oranges Well roasted, with sugar and wine in a cup, They'll make a sweet Bishop when gentlefolks sup. ~ LAWN SLEEVES, CARDINAL, AND POPE, Owe their origin to some Brasen-nose Bacchanalians, and differ only from Bishop, as the species from the genus. LAWN SLEEVES. Substitute madeira or sherry for port wine, and add three glasses of calvesfeet jelly. CARDINAL. Substitute claret for port wine; in other respects the same as Bishop. POPE. Precisely the same as Bishiop, with the exception of champagne being used instead of port wine.