2. Press “E” to rotate a face
Gotcha. Here we have our rascal 
Don’t press press E to use the widget. Just click with the mouse at the handler and pull 
Pressing E, or W or R enters the manual transform mode. That’s an independant mode* from the widget. The useage of the Widget is simply grabbing the handler with the mouse and pull, without holding any hotkey.
* It offers you for example fine control with holding down shift. Something that the widget does not offer. And it is not bound at axis. At least not as long as you don’t limit it to axis.
In Blender, it seems like there’s an extra step I keep missing.
In Blender this extra step is to manually change the widget type. We might hotkey this at one point. There is an idea to connect it with the WER keys too, with holding control or alt or shift. But currently you have to click at the widget type in the header that you want to choose-. See Image above, Move, Rotate, Scale.
Hope that helps
Reiner