fix(app-vite): audit BEX mode and guide

[hawkeye64]

Summary

• keep BEX assets, locales, and Firefox public files synchronized during development, including file removals
• align bridge examples and type comments with the current object-based messaging API
• document the existing bex.minify option and modernize BEX setup, debugging, and TypeScript guidance
• correct copied text, terminology, links, and manifest examples throughout the BEX guide

Manifest JSON examples

The inline // Popup Page, // Options Page, and // Dev Tools labels were removed from the manifest examples because browser-extension manifests must be valid JSON, and JSON does not support comments. The property names already identify each entry, while removing the comments keeps the examples directly copyable into manifest.json.

Root causes and impact

The development watchers copied changed files but did not remove stale generated assets. The Firefox public-file watcher also copied changes beside manifest.json rather than into the generated www directory. This could leave the loaded development extension out of sync with its source. Several guide examples still used the former positional bridge API and listener response shape, causing copied examples to fail against the current implementation.

Validation

• syntax checks for the modified BEX runtime files
• formatting and lint checks for the affected code, types, and documentation
• git diff --check
• direct asset synchronization verification covering initial copies and source deletions
• repository pre-commit formatting and lint checks