Royal Artillery / Royal Engineer Colar Dogs

[AIRCOP]

I recently received a nice box of badges and buttons from Bermuda. In the lot were three "UBIQUE" collar dogs (see attached picture). In trying to identify these, I found that the RA / RE possibly wear the same insignia? Is this correct? If not how do you differentiate between RA/RE badges?

Another observation, two of the badges have what looks like a coil of rope on the grenade body. One does not. Why the difference?

Lastly, why two different badge sizes? Who wears what?