docs: Update manual publishing steps a little bit.

[ditman]

Description

Updates the manual publishing steps with what I needed to do to release stuff this week. This just expands the detail of the existing doc a little bit.

Pre-launch Checklist

• I read the Flutter Style Guide recently, and have followed its advice.
• I signed the CLA.
• I read the Contributors Guide.
• I have added sample code updates to the changelog.
• I updated/added relevant documentation (doc comments with ///).
• If my PR is a fork PR, I've checked that [e2e tests] passed.

If you need help, consider asking for advice on the #hackers-devrel channel on Discord.

[gemini-code-assist]

Code Review

This pull request updates the manual publishing documentation in docs/contributing/publishing.md to provide detailed, step-by-step instructions for releasing packages. The feedback suggests enhancing these steps by explicitly adding commands to switch to the main branch and pull the latest changes, as well as recommending a dry-run publish step (flutter pub publish --dry-run) to catch potential issues early.

_{Note: Security Review has been skipped due to the limited scope of the PR.}