What does 'Astra' here mean? Surely they must be referring to the Latin word.

Because in another dead language of antiquity, Sanskrit, it means "weapon". Which would be a bit too on-the-nose.

Citing Tibo [0]: "

- Bigger number = Better

- Bigger celestial object = Better

and the scale is Astra > Sol > Terra > Luna. "

[0]: https://x.com/thsottiaux/status/2095600295808283073

It's clearly an extension of the previous naming: Luna, terra, sol

Luna, Terra, Sol, Astra. Though Sun is also a star, should have called it Galaxy or something.

Quite clearly in the same vein as Sol, Terra, Luna.