Fix for #6171

[JAPNITSINGH]

• Added live indicator when run or debug flutter app in the tool window tab.

---

• I’ve reviewed the contributor guide and applied the relevant portions to this PR.

Contribution guidelines:

• See our [contributor guide](https://github.com/dart-lang/sdk/blob/main/CONTRIBUTING.md for general expectations for PRs.
• Larger or significant changes should be discussed in an issue before creating a PR.
• Dart contributions to our repos should follow the Dart style guide and use dart format.
• Java and Kotlin contributions should strive to follow Java and Kotlin best practices (discussion).

[JAPNITSINGH]

I have marked this as draft for now. I am planning to create a ToolWindowHelper (or use an existing one) in order to move the code away from LaunchState.java file.

[JAPNITSINGH]

I have moved the code to ToolWindowBadgeUpdater.java to handle the live icon indicator. The issue can be found here

[jwren]

Thank you for the PR! Please add a screenshot showing the change in the UI.

[JAPNITSINGH]

@jwren Here are the screen shots after my code changes.

1. Run button in the tool window:
   

2. Debug button in the tool window: