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This Limited Admin scenario allows a user to be able to run a preconfigured job, or both run and edit it. In this case latter, the account's VFS has to point to the top level directory the job operates on, or the Allow viewing of server files, including the server log, job logs, and any other file on the server... right granted.
First need to set up the user account with the proper permissions

Then add the user name to the Users allowed to monitor list on the Job's Setup page.

Such a limited admin account will only be able to see, and if the role permissions allow, edit the jobs designated to him.