docs: expand authentication examples

[scottaddie]

Why

Improve the authentication guide's cross-language coverage and make the examples easier to compare.

What changed

• Add Rust examples for signed-in users, GitHub OAuth Apps, and disabling auto-login.
• Add Go, Rust, .NET, and Java examples to the environment-variable section.
• Order each language tab group alphabetically and use TypeScript as the tab label.
• Use language-neutral guidance for client token configuration and logged-in-user fallback.
• Link the GitHub OAuth App section to the detailed OAuth setup guide.

Validation

• TypeScript documentation validation passed.
• Go documentation validation passed.
• C# documentation validation passed.
• Full validation is environment-limited because Python and Maven are unavailable.

[Copilot]

Pull request overview

Expands authentication guidance with consistent cross-language examples.

Changes:

• Adds Rust authentication examples.
• Expands environment-variable examples across SDKs.
• Standardizes language tabs and OAuth guidance.

Show a summary per file

File	Description
docs/auth/authenticate.md	Expands and reorganizes authentication examples.

Review details

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• Files reviewed: 1/1 changed files
• Comments generated: 1
• Review effort level: Balanced