I've just tried doing it again from scratch. I copy-pasted your formula from above (the {IF {INCLUDEPICTURE {IF TRUE "C:\Users\Elena\Pictures\{MERGEFIELD Picture}"} \d} {INCLUDEPICTURE {IF TRUE "C:\Users\Elena\Pictures\{MERGEFIELD Picture}"} \d}} variant) and swapped in my fields. And replaced the brackets with Ctrl-F9 of course.

When emailed, the picture is still linked. Viewed with Alt-F9, it displays an INCLUDEPICTURE with the link to the file ending in \* MERGEFORMATINET. If the file is not where expected, it displays the standard "linked picture cannot be displayed" message and red-cross image.

I tested it side-by-side with my original - the two look to be identical and behave in exactly the same way.

I can't copy-paste it because Word doesn't let me do that with field codes. I've uploaded a screenshot; annoyingly the forum rejects the URL for some reason, but it's here for 28 days: http://www.tiikoni.com/tis/view/?id=be05946

The resulting code in the saved, emailed document is here:
http://www.tiikoni.com/tis/view/?id=4c62eb0

Since the image cannot remain linked if the code is correct, can you suggest what is wrong with the code? I'm stuck. The only discrepancy I can think of is that the image is on a shared drive (M rather than on the hard drive. However, that's not an option.