JIRA custom fields can have their default values and possible options differ for each project/issue type. The JIRA user interface seems confused about the terminology of these things. Here is a screenshot of the custom field page, with two of these things configured:

The problem lies with the "Edit configuration" link. It should be renamed to "Edit context", since:

• The label on the left says "Applicable contexts for scheme".
• The page you get to after clicking is titled "Modify configuration scheme context, and is indeed all about setting the context.
• You can't edit the custom field options on this page, which are certainly part of the context.

So really you're editing a context, not a configuration.