What is the name ideavim action-name of "stretch editor tabs" ?

You can assign any shortcut in Settings: 

I already tried that, but it doesn't work. I even made sure that my hotkey for it wasn't clashing with any other hotkey. I thought it might because I had ideavim enabled, and that I might solve it by making an ideavim k trigger the corresponding action. 
I dont know ... any idea why it is not working? 
I split the editor windows into multiple windows, place my cursor in one of the windows and presses "ctrl + Left arrow" ... but nothing happens.

You are assigning a shortcut to a custom plugin's action (Tab Shifter). It could have compatibility issues with the vim, or some specifics of your installation. 

Try assigning the shortcut to "Stretch Editor to Right" IDE action (like on my screenshot), which is bundled one.