Some things I have noticed:
When I right-click on the image in the original document, the one that enlarges, I get this menu. Note that it has an item called "Presentation Object".

When I try inserting or embedding the new image I get a slightly different menu that contains an item called "Slide Object".

When I double-click my image, instead of opening in a full screen new window, it displays in a "cross-hatched" border, which indicates Word has opened it for editing (sorry, I can't reproduce the actual image outside of work so I have replaced it with some kittens). Click this image to enlarge it - because it works here!

There must be some kind of insert or embed option to produce this enlarge action, but I can't seem to find it. Even my google mojo has failed - I found one set of instructions that said it would enlarge the image, but following them carefully just gave me the edit option shown above.
Any help gratefully appreciated!