Refactor AI Toolkit documentation to update LLM provider references a…

[csells]

…nd improve clarity

• Updated references from deprecated google_generative_ai and vertexai_firebase packages to the new firebase_ai package across multiple files.
• Enhanced descriptions and examples in the chat-client-sample.md, custom-llm-providers.md, and feature-integration.md to reflect the latest changes in the AI Toolkit.
• Improved formatting and consistency in the documentation for better readability and user experience.

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• [x] If this PR is not meant to land until a future stable release, mark it as draft with an explanation.
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• [x] This PR uses semantic line breaks of 80 characters or fewer.

[gemini-code-assist[bot]]

Summary of Changes

Hello @csells, I'm Gemini Code Assist[^1]! I'm currently reviewing this pull request and will post my feedback shortly. In the meantime, here's a summary to help you and other reviewers quickly get up to speed!

This pull request refactors the AI Toolkit documentation to align with the new firebase_ai package, replacing deprecated LLM provider references and unifying configuration instructions. It also significantly enhances clarity, improves formatting, and introduces new sections for advanced features like function calling and custom speech-to-text, ensuring the documentation accurately reflects the latest SDK changes and provides a better user experience.

Highlights

• LLM Provider Migration: Updated all documentation references from the deprecated google_generative_ai and vertexai_firebase packages to the new firebase_ai package, ensuring consistency with the latest SDK.
• Documentation Clarity and Formatting: Enhanced descriptions and examples across chat-client-sample.md, custom-llm-providers.md, and feature-integration.md for improved understanding and readability, alongside general formatting and consistency improvements.
• New Feature Documentation: Added new sections in feature-integration.md detailing 'Function calling' and 'Custom speech-to-text' capabilities within the AI Toolkit.
• Simplified Configuration: Streamlined the configuration instructions in index.md to reflect the unified firebase_ai package, removing separate setup guides for Gemini API keys and Vertex AI.
• Enhanced User Experience Details: Updated details on multiline text input gestures for Web/Desktop and introduced an 'Attach Link' option for multimedia input in user-experience.md.

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[csells]

fyi: @mit-mit @sfshaza2 new draft PR with consistent branding changes to Firebase AI Logic SDK as well as a much cleaner set of files (the other PR went a little whacko on me).

[flutter-website-bot]

Visit the preview URL for this PR (updated for commit 1bca581e9cb02908a675039d97141fcbb8a8948b):

https://flutter-docs-prod--pr12787-ai-toolkit-updates-2-d8r32a9k.web.app

[sfshaza2]

@csells, the build is failing because ai-toolkit/custom-llm-providers/index.html doesn't exist.

[csells]

I'm away from my computer today but custom LLM providers are discussed in the custom-llm-providers.md in the same folder. I can fix this tomorrow.

[csells]

I found the problem. The recommended:

dart run dash_site check-all

doesn't check all, e.g. it doesn't check link references. The readme doesn't say to do this, too:

dart run dash_site build
dart run dash_site check-link-references

to check references.

Reference error found and fixed.