The debate whether "batting around" involves sending nine or ten hitters to the plate in one inning is one that has been around since... I don't know, but I'm guessing some time in the nineteenth century. For some reason the topic is still hotly debated every time a team sends nine guys up to bat in a frame or even comes close, and flares up on a grand scale once or twice a year. The most recent hubbub came a few days ago, and I got absolutely nothing accomplished at work that afternoon thanks to devoting all of my energy into poking holes in nonsensical overthinking.
To me, it all hinges on the word "around". Nine completed plate appearances brings you back around to where you started. The circuit is complete. To this day I have never understood how any more explanation than that could possibly be needed, but some people cling to their ten theory the way conspiracy theorists claim the same "extra" shows up in the background of footage of tragedies, thus proving that everything is a hoax and the rest of us are just sheeple who need to open our eyes.
One example I like to use is that a day is from 12:00 to 12:00. I have always disliked the phrase "the eleventh hour" to refer to the last possible opportunity to do something, because the hour beginning at 11:00 that they're referring to is actually the twelfth hour, and that seems applicable here. Once that twelfth - or twenty-fourth - hour ends, as soon as 11:59:59 rolls over to 12:00, you have gone around the clock. You don't wait until 1:00 or 12:01 or even 12:00:01 to declare it a new day. Where one ends, another begins; there is no overlap.
As I was patiently explaining this to someone I drew up a visual aid. It isn't the tidiest, as I threw it together in a hurry at my desk, but it gets the point across.
Jodes also had a great example that I liked even more than my own. She stated that in order for someone to score a run, they go from home plate back around to home. Saying one batter has to come to the plate twice in an inning to constitute batting around would be akin to saying that a runner has to circle the bases and then run to first base a second time in order for a run to count.
I've heard probably every possible argument for the ten-batter cause, and they all hinge on disregarding the definition of one of the two words: around. While I can appreciate the thought that goes into some of the reasoning, none of it holds up.