Profiler and Visual Profiler start/stop state inconsistency

[lostminds]

Godot version

4.0.beta8

System information

macOS 13.0.1

Issue description

This may be borderline usability improvement, but I think it can be considered a bug report. Basically the bug here is that the Visual profiler, unlike the Profiler is not automatically stopped when running the game if it's been left running from the previous run. So, the Visual profiler appears to be running but it is probably not, or at least not connected to the current running game instance.

Beside the simple reported bug here that the Profiler and Visual Profiler behave differently (and theProfiler slightly better) a more robust usability improvement to fix this issue as well could be to stop both profilers when the game is quit, and possibly not allow starting them when the project isn't running to more clearly communicate that they only work while the game is running (see #65209).

Steps to reproduce

• Run a project (any project)
• Start the Visual Profiler in the Debug tab
• Stop the game
• Observer the profiler looks like it's still running
• Run the game again
• Observer the profiler looks like it's still running

Minimal reproduction project

Any project as it's an issue in the editor profiler interface