The problem
Start with an article that contains a maintenance template of some sort. Your goal is to fix the problem identified by the template. You want to end up with a better article and no maintenance template.

Open the article in the visual editor.

Click on the template.

Cool, there's another "edit" button. Editing to solve this problem is exactly what you want to do!

Now you're in the template dialog box. In this case, 'edit' doesn't mean that you get to edit the article to solve this problem. It means editing the template itself.

How to reduce confusion
Maybe there should be a "remove this template" button? Maybe "edit" should say "edit this template"? Maybe something else?