To be offered in the email editor, the template must also have the 'Use template in Email dialog' check box selected. The check box can be found on the 'Object type' tab of the email template editor.
On the other hand, the check box must be cleared in a template for sending emails that are not supposed to be edited manually, e.g. templates for automatically sent status change emails. These templates are used "in the background" but are not offered for selection in the editor.