After considering the matter for some time, the Board voted that the ground taken in the St. Louis protest was valid, and that the games due from the Mutuals and Athletic Clubs to the other six clubs should be reckoned as forfeited, and the championship was accordingly awarded on the basis of the following table of games lost and won.
-St. Louis Globe-Democrat, December 8, 1876
The final standings, as decided by the League, are a bit different than the one presented at B-Ref. Also, shouldn't Cincinnati and St. Louis have played one more game against each other? I assume their tenth game was a rainout or something that was never made up. I probably have that information somewhere but it's not all that important. Either St. Louis would have won and finished five back of Chicago or they would have thrown the game and finished six back. That's a joke, by the way.
Bottom line: St. Louis finished second in 1876.