No, but I think you're missing something. When you turn on
Show all filename extensions in the Finder, it actually affects more than the Finder. It affects the file names that appear in the title bar of documents, and it also therefore affects what AppleScript regards as the name of a document.
So the correct answer is to use the name of the document, just as it is when scripting any application. If the window displays an extension, you include it, otherwise you don't.
There's nothing unique to Script Debugger about this; it's the same for all applications. Try turning off extension display and address a document with a hidden extension in any app, and you will see it fail if you include the extension, and vice-versa.
The confusion probably stems from the fact that the rules for script libraries and applications are different. But in this case it's standard AppleScript document scripting, and therefore the standard rules apply.
Unfortunately AppleScript hasn't helped matters by telling you that the document doesn't understand the "testing" message, when the real error is that you have addressed a document that doesn't exist. You should log a bug with Apple about that.
So yes, there's an assumption that the user understands how AppleScript treats document names, and if you've always had extensions showing and never shared scripts with users who don't, you may never have come across this point.