The third piece of every template is the thumbnail image that appears in the +New menu when users create a new file.
There are two ways to set this image. One is via Setup > Templates, where you enter the URL of an image in the template settings. The other is by uploading an image to your template folder (most implementations use /_resources/ou/templates) and giving it the same name as the TCF it should be paired to.
Acceptable file formats for thumbnail images are
If your implementation has the "Template Location" set to Local (staging server) in Setup > Sites, then thumbnail images do not need to be published. However, all users will need access to the thumbnail image in order for the image to appear unbroken in the +New menu.
If the "Template Location" is set to Remote (production server) in Setup >Sites, then thumbnail images do need to be published.
Lower level users need access to local thumbnail images to avoid broken images in the +New page menu.