I was expecting Streets & Trips to associate the "no-fly" zones with a particular saved trip, but I don't think it works that way
Yes, it should work that way. If you open a new map (ctrl+n), the "avoid area" will no longer be present (unless you previously saved it to your default template).
If you wish to make certain you will not experience the same "mistake" during your travels again, you may change the color of your "avoid areas" to red, or some other color of your choice. That way the "no-fly" zone will be more difficult to overlook.
To change the color of an "avoid area", click on it to select it, and then select the desired fill color on the drawing toolbar. You may then right-click the avoid area again, and from the menu that opens up you may select to "Send behind roads".