🧹 [code health] Extract chrome.tabs listener callbacks

[savvides]

🎯 What: Extracted nested anonymous callbacks in chrome.tabs.onActivated and chrome.tabs.onUpdated event listeners into named functions (handleTabActivated and handleTabUpdated) in extension/sidepanel/sidepanel.js.
💡 Why: Reduces deep nesting and improves the readability and maintainability of the event listener setup logic.
✅ Verification: Verified by running the test suite (./test/integration-test.sh and ./test/smoke-test.sh). The refactoring preserves all existing functionality exactly as it was.
✨ Result: Cleaner, easier to read codebase with appropriately extracted callback functions.

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PR created automatically by Jules for task 5445234150967915865 started by @savvides

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