Yep, you are correct. You need the template to use the generic names, not the style specific names. Sorry about that. So it would need to replace the Journal field, with the words “Secondary Title” in the template. – While you have 6 Ref types, you could use the this method for your most common ref types, and then manually insert the others with the Ctrl+K shortcut.
Ref Type specific citations have been requested of the Developers in the past, but not with enough customer base support I guess.