Most recent birth, my water did not break. The MW finally got through it AFTER DD's entire head was born, as she was rotating to get her shoulders out.
Urge to push? I had an urge to NOT push, but my uterus wouldn't listen to my puny brain at that point. Instead of contractions that I pushed with (like in DD1's birth) I had one giant non-stop contraction (20-30 min pushing stage). I was trying as hard as I could to relax and not bear down, but any time my concentration was interrupted, my body pushed like crazy.
It was much much more intense than DD1's pushing stage with waters broken, and if there is a next time, I might ask for AROM at the start of pushing if we are still intact at that point.
ETA: That makes it sound painful, but it wasn't really. It was intense, and I felt like I was a bit out of control. I couldn't do anything except hang on tight to my husband's legs, but I wouldn't describe the sensation as pain.