Now that I can modify the fields in the the New Request form and Edit form
separately (using Ishai Sagi's Sharepoint Tips Utility Pack), I have another
problem.

The basic set of Help Desk site permissions is not adequate for rolling this out
to the general public (employees). So, I created a new custom permission based
on Contribute, but without Delete privileges. But, this still isn't
satisfactory. By allowing users to create new requests and edit their existing
requests, they are also able to view and edit requests created by and owned by
other people. They are also able to view the entire list of service requests.

If anyone else is, or has, dealt with this issue, I'd like to hear how you
resolved the problem.