I looked but didn't find more damaged pearls; I must have thrown them out already.
It is not unusual for people who don't know pearls to use the term "natural" to mean "not fake." However, when it comes to pearls, "natural" refers specifically to wild pearls, as opposed to farmed pearls (cultured pearls).
Your pearls have colors and shapes found in baroque cultured freshwater pearls. However, I am not by any means an expert on natural pearls. I look forward to reading other opinions.