Which one is more appropriate in the present continuous? She is having a baby or she is giving birth to a baby. Both are correct, but mean different things. "She is giving birth to a baby" means she is actually giving birth at this instant. She is in labour and about to/in the process of giving birth. "She is having a baby" simply means she is pregnant. It does not mean the actual birth process has started. You could say, "She is in hospital having a baby", but that only means she is ABOUT TO give birth, but it has not started yet.

The second sentence is the more appropriate one. It focusses on what someone did and we directly think of all the circumstances and hardships someone has. The first sentence is very casual. "She had a baby yesterday" is pretty replaceable by "She had a hamburger yesterday".