I know you asked about American English but for info, I think the language used in cricket is different to that used in baseball, despite their similarities as sports. In cricket, we don't call someone a "batter", it's always "batsman".
"Batsman" remains the most common term, especially in writing and TV/radio commentary on the traditionally preeminent forms of the game, international test matches, state and county competitions etc, but "batter" is now frequently heard in conversation and interviews and occasionally seen in newspaper and online reports. Probably more often in relation to shortened forms of the game like one-day internationals, T20 etc, but it's by no means confined to them.
Aust. captain Michael Clarke: ''I see overs from a lot of the batters as important as the overs you get out of your frontline quicks".
England off-spinner Graeme Swann: "I think most of our problems were mental. Maybe we didn't adapt – and not just the batters, the whole team".
With the increasing profile of international women's cricket, "batter" has almost taken over in that context from the surprisingly resilient "batsman", which is nevertheless still quite common.
Wikipedia: "The most notable is pro
bably Nadine George, a wicket-keeper/batsman, who became the first, and to date only, West Indian woman to score a Test century, in Karachi, Pakistan in 2003–04".
But this, from The Guardian (2009), is more the norm: "Claire Taylor, by showing herself to be a pugnacious batter with a fine cricket brain, has served as an excellent advert for women's cricket".
not a teacher