I'm pretty sure this is related to Vercel moving away from Vercel Postgres powered by Neon and instead offering multiple different db integrations through their new marketplace. Here's a flag from the docs:

Vercel Postgres is available on Hobby and Pro plans for customers with existing Vercel Postgres storage. You can create a new Postgres storage with the Neon Marketplace integration if you don't have an existing Vercel Postgres store. Existing Vercel Postgres storage will be automatically transitioned to the Neon Serverless Postgres product integration from the first week of December 2024, and expected to be completed by the end of December 2024. No action is required on your end and there will be no downtime and pricing change.

And then here's the migration guide from Neon's docs: https://neon.tech/docs/guides/vercel-postgres-transition-guide

My guess is the team just hasn't updated the tutorials yet to reflect this

You should either See Postgres or Neon depending on the account setup. @neversitdull is right, our docs are yet to be updated.

You can opt for Neon if you don't see Postgres there.