Hi there...congrats on your pregnancy!
What you describe is called a 'cervical lip'. As far as common, well, with directed pushing flat on one's back, it can be common, yes. But that kinda opens up a whole host of why active birth management is crummy and I'll try to stay away from that.
Vaginal exams to diagnose dilation and station are associated with creating cervical lips.
I'll tell you that I birthed without ever having anyone feel me besides my DH holding my hands and I will never, ever know if I got to 10 cm, if I was 'fully effaced', what my baby's station was, etc. But here he is, peacefully sleeping at almost a year!
My point is that I let my body push him out and it worked really well for both of us.
I've never heard any woman experience anything but a heck of alot of pain with a birth manager pushing aside tissue (pushing away the 'lip').
With induction, by definition, you are telling your body to birth (AROM, pit) without the baby initiating the process. Hence, there is a lot that can be difficult when that is the starting point. I am not judging you for an induction, just saying, that your experience is very common when someone ruptures membranes, breaks waters, induces strong contrax with the objective to push the baby out.
I would bet lots of money that if you were allowed to go into birth with your baby's initiation, allowed to have free movement in birth (no lying on your back with EFM), and be without vag exams, you would not experience a cervical lip again. I know that vag exams are thought to be harmless and information-yielding, but they come with risks such as cervical lips.