docs(keycast): document health probe ordering and shutdown semantics

[sergey3bv]

Summary

• Documenting the reason healthz/startup is safe without a DB check.

Avoid naming corporate partners, customers, or other sensitive external brands in this PR title or body. Use generic descriptors unless a maintainer explicitly approves the public reference.

Motivation

• A follow-up for Improve keycast health probes #93

Related Issue

• Closes docs(health): document health probe invariants and shutdown semantics #136

Testing

• cargo test --workspace --verbose
• cargo clippy --workspace --all-targets --all-features -- -D warnings -A deprecated
• cargo fmt --all -- --check
• Manual verification completed

Visuals

• UI/web change with screenshots/video attached
• No visual change
• Visuals and text avoid sensitive external brand or partner names unless explicitly approved