Whether or not you should be using a birth ball in the delivery room while giving birth is a hard question to answer. Every women is unique and different, every pregnancy and every birth has their own unique story, and what works for one woman may not work for another, just like what worked on your first birth may not work on the next one.
The stability ball / balance ball / exercise ball / labor ball / birth ball (gosh it has so many names! And all are correct, by the way!) is a wonderful tool that can help you go through the different stages of pregnancy and labor itself, in a calm and relaxed way. It can serve you for workouts, it can serve you when going into labor, and it even comes handy postpartum as little ones love it! The ball can be used during labor (I used it myself, twice ;)), however most women are not so sure what to do with it, which might lead to embarrassment or awkwardness of not knowing exactly what to do. I'm a true believer in gaining as many tools and knowledge as possible. Worst case scenario - You learn and then decide not to use it. Better than not learning, and then wanting to use it but not knowing how to, right?? The way I see it, it's kinda' like learning to drive a car, even if you're not sure you'll get one: If you then decide not to buy - okay, but at least you have the skill if you feel like it later, or if one day you need to drive your friend's car for whatever reason. You can't just wake up one day and decide to drive if you didn't practice beforehand. Same thing here :) Learn, know your stuff, decide later. Skills are always good to have. Oh and the funny things is - the moves here are so simple, basic and easy to follow, that it's really easy to learn! So no reason not to, right?
I recommend practicing the use of you labor ball on a weekly basis during pregnancy: The more you practice the more likely your body will know what to do on its own. Just like any other new skill: The more you do it the better you get at it, the less you have to think about it. That being said, it is important to mention that at the end of the day - it doesn’t really matter how you use it during labor, as long as it serves your needs at that moment, even in the most passive way. When you're in a busy room with doctors, midwives & nurses coming in and out and you want to focus on yourself and zone out for a bit, having that familiar piece that is all yours might just be all you need.
Personally, I probably gained more than hundreds of hours (if not thousands) of practice with the ball, and yet - when I got into labor and was in pain (hey, that's life ;)), at some point just wanted to let my body melt on the ball and stop repeating the exercises I'm so familiar with. And that's exactly what I did. So should you be using the birth ball at birth? The bottom line is: that's totally up to you. I say, don't decide now. Just hop on that ball and get to know it a bit better :) It will only do you good. And, your little one would love it, too!