What was the first case of a celebrity playing a fictionalized version of her/himself?

Baseball players in 1911, 1912, 1914, 1915, 1916, 1920, 1921, 1927, 1942

The Guide to United States Popular Culture (2001) by Ray Broadus Browne and Pat Browne says:

As Linda K. Fuller has noted, "The real story of the success of baseball films is not the batting box, but at the box office. From the beginning, filmmakers were quick to realize the added box-office potential of the day's baseball stars, so it wasn't long before some of the began appearing, usually as themselves, in dramatized films as well as in newsreels." Ever since Hal Chase played himself in a 1911 film [Hal Chase's Home Run], followed the next year by Chief Bender, Jack Coombs, and Rube Olding in The Baseball Bug, others joined the film fun: Christy Mathewson in Love and Baseball (1914), Mike Donlin and John McGraw in Right Off the Bat (1915), Ty Cobb in Somewhere in Georgia (1916), Frank Baker in Home Run Baker's Double (1914), Jack Johnson in As the World Rolls On (1921), and Babe Ruth in Babe Comes Home (1927), as well as a number of shorts produced by Universal Pictures, like Fancy Curves, Just Pals, and Slide, Babe, Slide. Also, Roger Maris, Mickey Mantle, Ralph Houk, White Ford, and other actual players have made film appearances.

100 Things Yankees Fans Should Know & Do Before They Die (2012) by David Fischer says:

Ruth also played himself in 1920 in the film Heading Home, and he played a fictionalized version of himself seven years later in the movie The Babe Comes Home, as the character Babe Dugan. In retirement, Ruth played himself in the 1942 movie The Pride of the Yankees, based on Lou Gehrig's life.

But were these parts any more than just a cameo?

At least Hal Chase's role in Hal Chase's Home Run (1911) was more than a cameo:

... in which the leading role is played by the famous First Baseman and Manager of the New York American League Team...

Likewise, Frank "Home Run" Baker played the leading role in Home Run Baker's Double (1914). Here's an account of the 'photoplay', and it certainly sounds like a fictional story of a baseball player, his doppelganger and some crooks.

Love and Baseball (1914), according to the New York Times:

Legendary baseball player Christy Mathewson played "himself" in the Bison two-reeler Love and Baseball.

Here's a 1914 paper's account of the story.

Heading Home (1920) (watch online) stars Babe Ruth as a fictional version of himself. Wikipedia:

It attempts to create a mythology surrounding the life of baseball player Babe Ruth... Ruth stars in the film, playing himself, but the details of his life are completely fictionalized.

And from IMDb:

The "true story" of baseball great Babe Ruth; Ruth plays himself.
...
If this is the "true story" of Babe Ruth, then I am a Llama!

Cowboy gangsters in 1912, 1914, 1918

Materializing Democracy: Toward a Revitalized Cultural Politics (2002) by Russ Castronovo and Dana D. Nelson says:

During its first years of statehood, Oklahoma itself temporarily became a center for filming westerns; characteristically, given the wild-and-woolly reputation of the first whites who had come to Indian Territory, the peculiar western subgenre that developer there was the cowboy-gang narrative produced and performed by the ex-gangsters themselves: reformed train robber Al Jennings played himself in The Bank Robbery (1914), and Emmett Dalton (last survivor of the notorious Dalton gang) played himself in The Last Stand of the Dalton Boys (1912) and Beyond the Law (1918).

The earliest TV Show would probably be The Jack Benny Program ( from wiki: The television version of The Jack Benny Program ran from October 28, 1950) Jack Benny played himself although it was a caricature much like James Van Der Beek. There may be earlier examples though from movies.

On television, a couple of early examples (though not necessarily the earliest) might be The George Burns and Gracie Allen Show or The Adventures of Ozzie and Harriet, although it's debatable if those are really fictional versions of themselves or just characters with the same names. Then there is The Monkees in 1962, which is most certainly them playing themselves.

As far as film goes, others might be able to chime in better, but you might be able to consider Laurel and Hardy as often playing themselves (though it depends on the film, of course).