fix(server): support JACK_MANAGEMENT_KEY

[roziscoding]

Summary

• accept JACK_MANAGEMENT_KEY or the legacy MANAGEMENT_KEY on the server
• prefer JACK_MANAGEMENT_KEY when both variables are defined
• update Compose examples to use the prefixed name
• audit repository documentation against the current code and fix stale routes, defaults, environment variables, peering guidance, and UI auth details

Verification

• bun test apps packages — 422 passed
• mise run lint
• mise run typecheck
• docker compose config for both example Compose files
• git diff --check
• independent pre-commit review: passed

Greptile Summary

This PR adds backend support for the preferred JACK_MANAGEMENT_KEY environment variable while retaining the legacy alias.

• Prefers JACK_MANAGEMENT_KEY when both key names are supplied and adds precedence/compatibility tests.
• Updates Compose examples to pass the prefixed key consistently to the backend and UI.
• Corrects documentation for management authentication, routes, configuration defaults, peering, logging, and retry behavior.

Confidence Score: 5/5

The PR appears safe to merge with no concrete changed-code defects identified.

The prefixed key is mapped into the existing nonempty validated management-key field, the legacy name remains supported, precedence is intentional and tested, and the deployment examples provide the same required value to both services.

_{Reviews (1): Last reviewed commit: "fix(server): support prefixed management..." | Re-trigger Greptile}