That said, this form of the game probably did not resemble modern baseball except in the vaguest way (i.e.: a bat, a ball, fielders, outs), and varied wildly across place and time.
when he stumbled upon an early game being played at the Elysian Fields in Hoboken, New Jersey in the 1850s, at which point he fell in love with baseball.
Americans have been playing bat-and-ball games with the name "base ball" or suchlike since at least the 1790s: a document from 1791 was found in the city of Pittsfield, Massachusetts, that mentions "base ball" (specifically in the context of it being banned anywhere in the vicinity of the town hall's expensive glass windows).
You will notice that these nearly-modern rules coalesced in the 1850s.
You might also recall that The American Civil War began in 1861.
Thus, over the course of the war, the New York baseball of the 1850s became America's baseball.