chore(firebaseai): merge same parse logic from developer api and vertexai

[cynthiajoan]

Description

Replace this paragraph with a description of what this PR is doing. If you're modifying existing behavior, describe the existing behavior, how this PR is changing it, and what motivated the change.

Related Issues

Replace this paragraph with a list of issues related to this PR from the issue database. Indicate, which of these issues are resolved or fixed by this PR. Note that you'll have to prefix the issue numbers with flutter/flutter#.

Checklist

Before you create this PR confirm that it meets all requirements listed below by checking the relevant checkboxes ([x]).
This will ensure a smooth and quick review process. Updating the pubspec.yaml and changelogs is not required.

• I read the Contributor Guide and followed the process outlined there for submitting PRs.
• My PR includes unit or integration tests for all changed/updated/fixed behaviors (See Contributor Guide).
• All existing and new tests are passing.
• I updated/added relevant documentation (doc comments with ///).
• The analyzer (melos run analyze) does not report any problems on my PR.
• I read and followed the Flutter Style Guide.
• I signed the CLA.
• I am willing to follow-up on review comments in a timely manner.

Breaking Change

Does your PR require plugin users to manually update their apps to accommodate your change?

• Yes, this is a breaking change.
• No, this is not a breaking change.

[natebosch]

Reusing more of the vertex parsing for the developer API looks like it should work out OK. I think as long as we aren't passing arguments like developerParseMode: true it makes sense to merge code paths. Any place where we would need a conditional I think we should continue to lean towards disparate implementations.

[Lyokone]

A couple of comments to review,