chore: upgrade prettier to v3 in website/playground

[DerTimonius]

Summary

This PR will upgrade prettier to currently 3.0.3 and fix problems associated with the upgrade:

• function is now async
• using prettier.format instead of debug.printToDocString.formatted
• add the pluginEstree plugin like mentioned in this prettier issue
• @types/prettier has been removed, types are now part of prettier
• type PrettierOptions had to be extracted to the types.ts file

Test Plan

I could not find any e2e Tests for the website/playground, maybe some should be added either with this PR or in a separate PR?
From clicking through the playground you can see that it's still behaving as expected

closes #302

[Conaclos]

Thanks! :)

I could not find any e2e Tests for the website/playground, maybe some should be added either with this PR or in a separate PR?

What do you propose?

[DerTimonius]

Playwright comes to mind, I have some experience writing e2e tests with it. Maybe it's an option to check if both the biome and prettier output are behaving as expected