Here's how Moon Starlight's birth goes in Night Furies vs. The Nightmare Family Part 2.
[We see Luna on a mederate]
Princess Luna: [panting]
Princess Celetia; You're doing just fine, Luna.
Nurse: Alright, your highnees! Push!
Princess Luna: [groans as she does]
Princess Celestia: [sees it] There's the head!
Princess Luna: [goes again]
Foal: [starts crying]
Nurse: Great job!
Doctor: It's a girl!
Princess Luna: Could I see her?
[the doctor gives her new foal]
Princess Celestia: She's beautiful, Luna.
Princess Luna: I know.
[then Yuna and Snowdrop come in]
Yuna: Wow, she almost looks like me!
Princess Luna: Yeah.
Hiro: What shall you name her?
Yuna: How about "Moon Starlight".
Princess Luna: That's a nice name, sweetie. "Moon Starlight".
Moon Starlight: [opens her eyes]
[The episode ends]