Agreed, BH is more like the entire tummy feels tight and hard. It's not usually painful. When I first felt it, I just thought baby was pressing/stretching out. Now my stomach feels like a bowling ball during BH 😅 I think lightning crotch varies for people. For me, if I bend down at the wrong time, or if baby moves in a weird way, it feels like pain in my cervix almost. It's a sudden sharp pain that goes away quite quickly. What you're describing almost sounds like nerve pain, since it travels down your leg. I doubt it's anything to worry about, but certainly worth mentioning to your Doctor. Best of luck Mama ❤️
What you're describing sounds like sciatic nerve pain. Heat helps aleaviate some of the pain if it's bothering you
Braxton hicks is more on your tummy that feels stiff/firm for seconds. Lighting croctch on the other hand can be felt near pubic area I believe.