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Why do a lot of applications provide an exit button in addition to window title one and Alt+F4?
I can understand it if:
What about rest of applications?
I'm not sure if I fully understand your problem, but one of the main tenets of creating good design is redundancy and accessibility.
Creating multiple paths to the same solution accounts for a wider audience, and caters to personal preference. In your example, not everyone knows about the Alt + F4 shortcut, hence the Exit button is the only way.
You can look at the Instagram app or Music Apps as an example. You can double tap an image, or click on the heart icon; you can press Next or swipe to the next song.
People have different needs, tools, and hardware. By providing solutions that utilize different input methods, you are creating a more accessible environment. In your example, is a keyboard/mouse is a common hardware configuration, but what about users that only use a mouse? Or only a keyboard? Or a touch screen? Or voice-input commands?
The close button in the top corner is provided by the operating system, and not the developer of the application. It will close the window even if the application developer does not handle whatever "on close" event the OS provides.
If an application consists of multiple windows, the close button will usually apply to the window rather than the entire application (although it is up to the developer how to handle this, and they can programmatically trigger the application to exit when a specific window closes). The program will usually be closed by the OS when there are no more windows open.
In the above scenario it would be clearer to many users if you just provide an "exit application" feature in the standard file menu location.
