Francesco, that's right.

Perhaps you can say where you saw the term "fake Calabi-Yau." I can think of several possibilities, but I can't guess without context.

...sorry, nothing seems to be working correctly. One more try: $p_*M_Y=F_1\oplus_{O_X^*} F_2$, where F_1 and F_2 are funny log structures whose ghost sheaf ($F_i/O_X^*$) is a copy of ${\mathbb N}$ on one of the two branches, but extended by zero over the origin. I'm still not sure if this is what you are looking for. I think the particular push-forward log structure you described in your original question is not a particularly well-behaved one, e.g., it is not fine.

See mathoverflow.net/questions/110615/difference-of-curve-classe‌​s/…

As Robert points out, it is easy to find examples where the holonomy group is not all of $SU(3)$, such as a product of a K3 surface and an elliptic curve, or quotients thereof where the metric is an orbifold metric. But there are still no compact examples with full $SU(3)$ holonomy. It would be nice to see further progress on this problem.