Provider update should let back end infer the project ID

[jhrozek]

Summary

We're able to infer which project ID to use if it's not specified
explicitly by inferring who the caller is and using their "root"
project.

Related: #3509

Change Type

Mark the type of change your PR introduces:

• Bug fix (resolves an issue without affecting existing features)
• Feature (adds new functionality without breaking changes)
• Breaking change (may impact existing functionalities or require documentation updates)
• Documentation (updates or additions to documentation)
• Refactoring or test improvements (no bug fixes or new functionality)

Testing

manually

Review Checklist:

• Reviewed my own code for quality and clarity.
• Added comments to complex or tricky code sections.
• Updated any affected documentation.
• Included tests that validate the fix or feature.
• Checked that related changes are merged.