Appropriate way to describe batter's count (grammar help)

Baseball layman here:
Would one say “Shohei Ohtani is now down 0-2 in the count” when he is batting and has no balls and two strikes, or “is now 0-2 in the count”, or use some other grammatical structure?

Either one of your examples work. Saying a batter is “down” in the count just signifies that the pitcher has the advantage based on the ball-strike count. You would never say “Ohtani is down 3-0 in the count” or even “down 1-1 in the count” but you can use the more netural phrasing o matter what the count is.

Both are fine.

Also common to phrase it as “oh and two” in speech, if that’s important to you, but the “and” can be omitted.

Ok thanks!

Thanks - now, also, if a pitcher balks, has he “committed” a balk, “pulled a balk,” or just “balked?”

“Committed a balk”, or “balked”. Or “that was a balk”.

Similarly, a “passed ball” (a pitch gets away from the catcher, but it’s the catcher’s fault, as opposed to a wild pitch, which is the pitcher’s fault) would be “he committed a passed ball” or “that’s a passed ball,” never “he passed the ball.”

And players never “pass” the ball to each other, as in other sports. They throw it (overhand) or flip it (underhand, for short throws). I hear kids say “Pass it to me” all the time and I correct them. It’s another insidious way that soccer is ruining American youth.