OK, after you've edited the task.eddmod.fm file, open task.edd.fm then click Edit > Update References to pull in the changes. Save task.edd. Then open task.template.fm and click File > Import > EDD and choose task.edd. Save the template.
(Of course backup the files before you change them in case something breaks. Also, you can either run FM as an administrator to be able to save the files directly or save them somewhere else and copy them back into the normal location.)
Now if you click File > New > DITA > task, you'll see that the context label is gone. These changes affect how dita files are displayed within FrameMaker. However they don't affect how anything is published - the next steps fixed the output for me:
Open composite.edd.fm (it's here: C:\Program Files\Adobe\Adobe FrameMaker 2019\Structure\xml\DITA\app\DITA-Composite-FM) find the context and result elements, and delete the prefix rules in the same way that you did in task.eddmod.fm. Save the file.
Open ChapterOutputTpl.fm and ditabase1.3.template.fm (or just ditabase.template.fm if you're using an earlier version of DITA). Use File > Import > EDD to import the composite.edd.fm into these two files. save them into their original locations. Now your published output should be good.
If you only care about the published output and not how things display when you're working with the base DITA files then you only need to change the composite, chapterOutput, and ditabase1.3, but I find it helpful to have things look consistent all the way through the editing process, so I make my changes in both places.