refactor(publish): modularize publishing logic

[Goku-kun]

This pull request refactors the initialization of external services and improves code organization in the publish directory. The main changes involve moving the setup of Firebase and MeiliSearch clients into their own modules, and renaming the Markdown parser file for clarity.

Refactoring and code organization:

• Moved Firebase initialization and Firestore export from index.js to a new module lib/firebase.js, and updated imports accordingly. [1] [2]
• Moved MeiliSearch client initialization from index.js to a new module lib/meilisearch.js, and updated imports accordingly. [1] [2]
• Renamed the Markdown parser file from parser.js to parsers/mdx-parser.js for better clarity and organization, and updated the import in index.js. [1] [2]