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!

Foal: [crying]

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]