I think the ending of your post was meant to be:

"In wysiwyg mode the image entries look like this:

[img param="569d584df3e2f-2.jpg"]http://res.cloudinary.com/mk1-golf-owners-club/image/upload/c_limit,w_1024/v1453188775/uploads/attachments/569de6a66be54.jpg[/img] [^]

In non wysiwyg mode the image entries look like this:

[attachment_safe description="569d584fdd167.jpg" thumb="0" type="inline"]10553[/attachment_safe]"

Because you're getting the 'img' tag in WYSIWYG, and the 'attachment' tag in non-WYSIWYG.

Or am I wrong and somehow within WYSIWYG you get both an 'img' and 'attachment' tag?

Also I'm a bit confused as we don't have any "responsive attachments" feature specifically. Possibly we have a max-width rule in the CSS or something like that.

This sounds like correct behaviour. If you add an attachment in WYSIWYG, it just is edited within the WYSIWYG (i.e. Composr allows it to convert to an 'img' tag). However, if WYSIWYG is off it just stores it as a pure attachment.