The earliest reference to baseball comes in a diary entry of an Englishman. Surrey solicitor, William Bray. He refers to playing 'base-ball' with his friends but as Scyld Berry points out the bigger claim that America's summer sport descends from the English game rounders does not 'get to first base'.
Nor has cricket always had it current unpopularity in the USA. The general contemporary view is that it is that Americans could never warm to a game played over five days and often ending in a draw. But in fact cricket 'was stronger in America than in Australia until the 1860s – stronger than anywhere else outside England'
More on the relationship between the rival summer games of 'two countries divided by a single language' here.
What is 'inside baseball'?
What is a curved ball?