I've heard from med school friends that apparently there were medieval contraptions that delivered the baby with the woman in a full squatting position. I've also heard stories from Buddhist texts of women giving birth to children in a standing position. There's even an organization that advocates giving birth in water. So my question may sound dumb, but I'm genuinely curious of different birthing methods and positions that have been advocated/practiced throughout history.
The modern way of giving birth - by lying down - is actually not a particularly effective method of doing it. It's used so that doctors can see what's going on more easily and have more room to work. Historically, however, the squatting position you described seems to be very common. Aside from the examples you mentioned, I know that the mesoamericans also advocated squatting births. Water births are a real thing that is sometimes used today, but I'm not sure if any ancient cultures ever used it extensively.
hi! if you don't find what you're looking for in this sub, you might consider cross-posting this question to /r/AskAnthropology, or even /r/AskScience