When selecting Task Template under Event Management, the fields shown in the Template field are different depending on the user role.

1. Log in as a user with the evt_mgmt_admin role.

2. Go to Event Management > Task Template, and click New.

3. Make sure the Incident table is selected.

4. Examine the drop-down options in the template.

   Note that the list for evt_admin contain only part of the fields of the incident.

This is operating as designed. Incident ACLs are defined out of the box to prevent templates for all fields except a white list.

Look at 
/sys_security_acl_list.do?sysparm_query=nameLIKEincident%5Eoperation%3De5a2b9c50a0a0bc5005dd4a3096ac37b 

The ACL name incident.* excludes all fields from the drop-down Template field, while the other ACLs listed include each individual field into the drop-down list. You can change this ACL to fit your needs.