Thank you Brenden and Jan for the quick response. Brenden, as you noted, I should be able to show or hide the underlying web part properties in the widget by setting the visibility of each one. In my case the container and container title properties always show for widgets, regardless of how I adjust the visibility of corresponding web part properties in the widget properties (see screenshot links in initial post). This is the part that is baffling me. Also, I'm not seeing a property definition with a label called "Widget container" in the widget Properties or the System Properties tabs. That also seems strange to me and just adds to my confusion. Where does the system get that text string for the UI?
Jan, you bring up a good question about why the Editable text web part is being exposed as a widget in the first place. Maybe it was never intended to be used in that way, and that's why I'm seeing this behavior. For this particular site that I'm working on, it may be too late to "unwidgetize" it. There is a lot of content built with it. Modifications were made to the site I am working on to give Editors more abilities to build layouts in the Page tab than normal. For example, the "web part zone" web part has been exposed as a widget to give Editors the ability to nest containers and influence the layout, without having access to the Design tab. I'm starting to think that exposing basic layout control to Editors in the Page tab was a mistake. What do you guys think?