Hello, the environment variable STEAM_FRAME_FORCE_CLOSE=1 from https://github.com/ValveSoftware/steam-for-linux/issues/1025#issuecomment-12860675 should give you the behavior you are looking for.
But it would be better if it would just do it without extra procedures. Steam client does not show at all and only the tray icon can be seen if a Steam game was started through the GNOME/Unity dash.
When that env variable was put into place in 2013, libappindicator was unreliable depending on which desktop environment you use. I do not know if the situation has changed, but my understanding is that it is undesirable to have steam is running without a user visible window and no tray icon at the same time.
If other apps can do it, why can't Steam fix theirs to do the same.
Examples of apps that can completely hide to tray: Transmission, Whatsie
Closing duplicate.
Your system information
Please describe your issue in as much detail as possible:
Most of the time I keep Steam open in the background while doing something else, the Steam icon remains in the taskbar and/or activities overview while having an icon in the system tray. It gets pretty annoying when you're doing something that requires constant switching of applications.
What is expected
When clicking the close button, it should act like the Windows version. It should hide the app from activities overview, alt-tab menu, taskbar and similar menus, while still being shown in the system tray.
Steps for reproducing this issue: