Pagat.com has an detailed overview of the game's history:
Original Poker, a game in which four players received five cards each from a 20-card pack and vied as to who held the best hand, evidently originated in the New Orleans some time between 1810 and 1825. Its gaming milieu was that of French-speaking maritime gambling saloons, especially those of the Mississippi steamers. Its name suggests that its first players felt they were continuing the tradition of playing a game called Poque in which one said Je poque to open the betting. At this time and place, and before it underwent development, Poque probably denoted a five-card vying game consisting of the central section of a formerly tripartite game of the same name.
...In the 1830s, having spread northwards along the Mississippi and westwards with the expanding frontier, Poker had adopted its anglicized name and become increasingly played with 52 cards to accommodate a greater number of players, thus also giving rise to the flush as an additionally recognized combination. Under the influence of Brag, its three-card British equivalent, it adopted the draw. This led to its further and more rapid expansion of popularity, as Poker-players preferred the additional round of betting after the possibility of improving a promising hand, while Brag-players preferred the wider range of combinations offered by a five-card hand.
British Brag certainly looks like an interesting game, with a similar style but somewhat different wagering and gameplay; American brag is often described as different from contemporary or modern English brag. Edmond Hoyle published his first treatise on whist in 1742, and his 1750 compendium of treatises described backgammon, brag, chess, pique, quadrille, and whist. These original volumes are apparently mostly held by collectors, and I can't find online copies. Hoyle's work became genericized and republished in increasingly-unrelated editions, which aren't on Project Gutenberg, but Google Books has some copies.
The American Hoyle (New York, 1864) has good descriptions of poker, brag, whist and faro as played in America at the close of the Civil War. The description of brag seems very similar to the English game, except the Knights and Nines are known as "braggers", which are ranked much like wild cards. However, vying appears to be much the same as in the English game.
This work describes poker as a straight game in which the players wager on the hand as dealt, without drawing. The cards are ranked from a pair, two pair, three of a kind, flush, full house, and four of a kind; it describes the straight as a regional variant. Draw poker, the historic "twenty-deck" or twenty-card poker, and (five-card) stud poker are all variants.
Other Hoyle-derived texts on Google Books include
- An Epitome of Hoyle (Dublin, 1791), which includes thirteen games, "with an account of the present fashionable Game, called E-O, played at most of the polite Chocolate Houses, never before attempted in print".
- Hoyle's Games Improved (London, 1814) includes essays on horse racing, skittles, and other games.
- Hoyle's improved edition of the rules for playing fashionable games (Philadelphia, 1838) includes a description of brag seems mostly similar to the English game, but makes clear that the Knaves and Nines (the "braggers") are wild cards. It's also one of the editions that contains a treatise on cock-fighting and breeding gamecocks.
- Hoyle's games (Boston, mid-19th century) describes billards, games for "Yankee-notion cards", and Thomas Frere's chess handbook (which includes a four-handed game on a Dessau board).
- Hoyle's games (New York, 1887) briefly describes poker as straight poker, without draw or stud variants or the straight. "Twenty-deck" poker, with a 20-card deck, remains a variant.
Dragoon Campaigns to the Rocky Mountains (1836) is the first printed reference to poker; it has a handy footnote to explain it as "a favorite game of cards in the south and west." Jonathan Harrington Green's 1843 anti-gambling screed describes it as a game for a 20-card deck "that is immensely destructiveperhaps more so than any other short game of cards now in use..." Joe Cowell's 1844 memoir relates a legend that it was invented by Henry Clay in his youth. Professional gambler and con man George Devol's 1887 memoir "Forty Years a Gambler on the Mississippi" is at both Google Books and Project Gutenberg.
Later, poker entered literature. The most significant description of a poker game in Mark Twain's Life on the Missippi is Chapter XXXVI, "The Professor's Yarn". It seems Stephen Crane's story "A Poker Game" was posthumously published in "New York City Sketches" in 1966. This book anthologizes some of the above, as well as 20th-century stories from folks like Bertolt Brecht, W. Somerset Maugham, John Updike, cryptologist Herbert O. Yardley and Truman's White House Counsel, Clark Clifford. McManus reports that Obama is the most recent poker-playing president.
Three-card poker is a modern poker-like game influenced by three-card brag, but without the distinctive vying element of either game, and a good house edge. A three-card straight poker could be an interesting toy poker, though, with fewer and simpler hand rankings.