Thank you for your post and details. The basic flow of how objects relate to each other is as such:
Project ---> Fields
By default TestRail will apply all fields to the default template. If you have fields that you do not want in your specific template, you will want to make sure that you edit each field and specify only the projects they should apply to as by default they will apply to all projects.
Templates will automatically apply and be available to all projects. When you create a test case you would simply make sure you are selecting the correct template.
Can you take a screenshot of the configuration one of the fields you do not want to show up and a screenshot of the settings for a field you do want to show up and attach them to this post? i.e., the projects they are set to show up in.