From TBIII: The 1884 Series was touted as "for the championship of the United States," but the influential weekly Sporting Life established precedent for future Series hype by naming victorious Providence "Champions of the World."
From the 1887 Spalding's Official Baseball Guide: In 1886 a series of games "for the world's championship" were arranged...
From the 1888 Spalding's Official Baseball Guide: Up to 1884 this question of what is now justly termed the "world's championship" in baseball remained an unsettled one;... In 1887 the world's championship series had become an established supplementary series of contests,...
From the 1887 Reach Guide: "Under the heading `THE GREAT WORLD'S SERIES' After the two championship seasons had closed..[St.Louis and Chicago]... agreed to play... to decide the "championship of the world," as it was termed.
The 1903 souvenir card referred to the World's Championship Games .
From the 1904 Reach Guide in the section "THE WORLD'S CHAMPIONSHIP SERIES": ... following is a complete resume of the world's championship series:...
The 1905 souvenir score card touted the World's Championship Series.
A 1905 New york program was entitled WORLD'S CHAMPIONSHIP 1905 / POST SERIES
A 1907 Official score cards for both Detroit and Chicago call it a World's Championship Series". This was repeated in 1908, by the Detroit program in 1909, and the Cubs program in 1910.
In G.H. Fleming's `The Unforgettable Season'
DETROIT- Sept. 19 (1908)- ... Last year Detroit owners gave their entire receipts to the champions, even after they had lost four straight games in the worlds's series.... - from the New York `American'
BUFFALO- Oct. 9 - ...Telegrams of congratulations poured in... one... from Barney Dreyfuss: "Hearty congratulations. Clean baseball was bound to triumph over affidavits and rowdyism. Best wishes for success in worlds's series." - from the Chicago `News'
The 1911 Shibe Park scorebook referred to the WORLD'S SERIES, as did the 1912 Boston, 1913-4 Philadelphia, 1914 Boston, 1915 Phillies & Red Sox, and 1916 Red Sox scorebooks.
From George Moreland's "Balldom" (1914) ...There was no world's series in 1904,... and refers to the "WORLD'S CHAMPIONS ATHLETICS" and under "WON WORLD'S TITLE IN 1910" : The Athletics were back in form again in 1910 and not only won the American League championship, but also defeated the Chicago Cubs for the world's championship,...
The 1919 Red's score card called them WORLD'S SERIES GAMES.
The 1920 Cleveland score card called it the World Series.
The 1921,1922, and 1923 Programs called it the WORLDS CHAMPIONSHIP SERIES.
In Ernest Lanigan's "The Baseball Cyclopedia" (1922) there is a section titled "World's Series' Facts and Figures from 1884 to 1921, inclusive." ... the World's Series today remains John T. Brush's.
The 1924 Washington score card said World Series 1924.
The 1925 Washington score card said World Championship Series.
The 1926 Cardinals score card said "for the Championship of the World".
The 1927-8 New York Programs called it the WORLDS CHAMPIONSHIP SERIES.
The 1929 Wrigley Field score card said WORLD'S SERIES, as did the 1930 Philadelphia card.
The 1931 Philadelphia score card simply said WORLD'S CHAMPIONSHIP .
The 1932 Wrigley Field and Yankee Stadium score cards said WORLD'S SERIES.
The 1933 score cards said WORLD'S SERIES.
The 1934 Detroit score card said WORLD'S SERIES, the 1934 St. Louis card said "for the Championship of the World".
In G.H. Fleming's "The Dizziest Season" Thursday, October 4 (1934) : The magic name of Dean, rather than any magic exhibited by Dizzy, beat the Detroit Tigers in the first game of the World Series. ...The term WORLD'S SERIES was also used on the score cards for: 1935 Chicago, 1935 Detroit, 1938 . The term WORLDS SERIES was also used on a score card for 1944. The Title WORLD SERIES was used on the score cards for : 1936, 1937, 1939, 1940, 1941, 1942, 1943, 1944, and ever program thereafter.
In summary, the name gradually changed
from the 1880's    "Champions of the World."
to the 1900's          worlds's series
(note the use of the uncapitalized plural possessive)
to the 1920's          World's Series
to the 1930's          World Series (modern form)