Who is the Real inventor of the baseball game
The short answer is that there is no single "inventor" of the game.
Abner Doubleday is traditionally credited with having "invented" baseball, in Cooperstown, NY, in 1839. However, it appears that a man named Alexander Cartwright was actually the inventor, in the respect that he first codified the rules of a game somewhat similar to today's game in 1845.
Doubleday himself never claimed credit for inventing baseball, or even for having been an important figure in its evolution. The only evidence to suggest Doubleday's involvement was a letter (written several decades later) from a man who would have been 5 years old in 1839. Even that letter did not suggest that Doubleday invented any of the modern elements of the game. Furthermore, the letter's author was insane.
In 1839, Doubleday was at the US Military Academy in West Point (he later served as an officer in the Mexican-American War, the Seminole Wars, and the Civil War, advancing to the rank of General in 1862), not in Cooperstown. Though he lived in Cooperstown before attending the Academy, his family moved away about the time he enrolled at the Academy, in 1838, so he would have had no reason to even visit Cooperstown in 1839. In any event, Cooperstown and West Point, though both in the State of New York, are separated by over 170 miles, a prohibitive distance to travel for a weekend, or even a whole week, of leave in those days.
But, to be perfectly honest, Cartwright's claim is little better than Doubleday's. It is, in fact, ridiculous to assign ONE inventor to the game. It evolved over a thousand years from games in England that bear little resemblance to the modern game of baseball, or even cricket (which, by the way, does not disprove the uniquely American nature of the current game). Sure, Cartwright wrote down the "Knickerbocker Rules", but teams had been playing under unwritten versions of those rules for years.
And, truth be told, even Cartwright's rules are a far cry from modern baseball. For example, the number of innings was not set (they played until one team reached a certain number of "aces", which we call runs now); there was no strike zone (batters had to swing and miss to get a strike); there were no home runs, except "inside-the-park homers" (any ball hit outside the field of play was a foul, even if it was over the outfield fence); foul balls never counted as strikes; the distance between the bases was not the same (and was not even standardized); the ball was pitched underhanded (by now you've probably realized that strikeouts were very, very rare); there was no "pitcher's mound", or any other place designated for the pitcher to pitch from (thus, presumably, the pitcher could stand anywhere on the field of play); a batted ball caught after the first bounce but before the second was an out (now it must be caught before the first bounce); the number of players on a team was not specified. Does that sound even remotely comparable to the modern game of baseball?
If you must assign an "inventor" to the modern game of baseball, then it has to be the MLB rules committee, and the date has to be the date (probably sometime in the last year) that the most recent of many, many official rules changes took effect. It is ridiculous to talk about some game played sometime in the past that was "essentially the same" as modern-day baseball. Thousands and thousands of minor changes add up to major changes, and the term "essentially" is way too open to interpretation.
If you want to talk about the first ever game that was called by the name "baseball", then you have to go back to at least 1744 (and completely give up any claim that it is an American invention) to a game that had very, very little resemblance even to Cartwright's rules, let alone modern Baseball.