Rebrand C# Health Check docs as C# Doctor

[Jake (JakeRadMSFT)]

C# Dev Kit’s approved user-facing experience name is now C# Doctor, and its Command Palette title is C#: Check health with C# Doctor. This updates the public documentation so first-time users see the same names as the product UI without turning every descriptive use of “health check” into branding.

Important

Release-order gate: Do not merge this pull request until the vs-green C# Doctor change !778912 ships. Public documentation must not introduce the new naming before the product. The docs are synchronized to the final vs-green product head f91bf0911534ea014336e3f5c6caf4df69d326d4; its extension content is byte-identical to built merge head 18e1f12ffb02d2adb5f6d2e16f1eb5040c1c5e4a. After !778912 lands, verify the landed product still uses the approved wording before merging.

Changes

• Define C# Doctor naturally as C# Dev Kit’s health-check view, then explain its tooling, runtime, stage, and recovery behavior.
• Use C# Doctor when naming the view or user experience and natural-language “health check” wording when describing the process or behavior.
• Update manual command instructions to say “Open C# Doctor” and “Run C#: Check health with C# Doctor.”
• Title the installation guide Install and manage .NET with C# Doctor without treating an unverified SDK installation phrase as an exact action label.
• Clarify the SDK recommendation and offline-workspace guidance.
• Update README navigation labels while preserving their destinations.
• Preserve legacy generated heading fragments with explicit anchors.
• Add a scoped README contribution rule for auditing these stable docs when the vs-green behavior or UX copy changes.

Source of truth and future updates

For maintainers with access, the vs-green implementation and approved UX copy remain the source of truth for the C# Doctor name, command, actions, states, setup, troubleshooting, and screenshots. A corresponding behavior or UX change should audit docs/health-check.md and the related guides in a linked documentation PR. The vs-green and documentation PRs should link to each other.

Compatibility

File paths, documentation URLs, relative internal links, link destinations, anchors, and aka.ms targets are unchanged. In particular, docs/health-check.md and every existing health-check* path remain in place. Explicit legacy anchors preserve fragments that would otherwise change with rebranded headings.

Redirect follow-up

None. This repository contains no health-check-related aka.ms alias or redirect configuration; the unrelated license and security aliases remain unchanged.

Editorial review

An independent editorial pass clarified the opening definition, installation-consent wording, SDK-recommendation heading, offline introduction, and maintainer freshness rule. The current guides contain no C# Doctor screenshots or related alt text to update.

Checks

• Verified all three manual Command Palette instructions use the exact approved C#: Check health with C# Doctor label.
• Verified old public labels no longer appear in repository documentation.
• Validated relative Markdown links and preserved legacy anchors.
• Verified the exact installation H1 and retained legacy fragment.
• Verified the committed diff has no whitespace errors.