Regarding the origin of this term as applied to baseball:

Cecil notes, correctly, that the term first appears in print in 1848, in reference to boxing. And he is of course correct in noting that baseball’s antecedents (rounders, cricket, etc.) would have had their batters facing east, and thus likely giving rise to the term.

However, according to Etymology Online, the word “baseball” (in reference to our game) dates from 1845, and no doubt was in circulation earlier. (Though the entry notes that earlier printed uses refer to rounders.) Thus, it is at least possible that the term was in use in baseball in the 1840s, before the boxing citation, but didn’t make it into print for some decades. (Not entirely implausible, given the disreputable perception of the sport at the time.)