Doc Revision: Custom UI Integration Section

[sandijean90]

Summary of Requested Changes:
Check for clarity, usability, and that the code snippets work on a fresh install and clean environment.

Requirements

• consistency with uv as the package manager
• code tested with a fresh install and environment
• language clear, descriptive, and matches language and tone of the rest of the documentation

Task Checklist

• Contributing Guideline Compliance: I read and complied by "Documentation Contribution Guidelines" section of the Agent Stack project CONTRIBUTING.md guide.
• Snippet Test: Verified all code works in a fresh environment (fresh agent stack install and clean environment).
• Visual Preview: Checked formatting/layout via Mintlify.
• CLI Sync: Ran mise run agentstack-cli:docs (if applicable).
• Location: Updates placed in docs/development for go live during next release or both docs/development and docs/stable for urgent go live.