Desktop manufacturing (DTM) - This was an early term and from which one of the early companies in the field derived its name (DTM, Inc.). Unfortunately, the only "desk" that one of these machines in the early days could have fit on was something like a Nimitz-class aircraft carrier. Only in recent years have any additive fabrication machines been able to fit on anything resembling furniture. There will very likely always be additive fabrication machines that won't fit on desks limiting the generality of the term. DTM doesn't come up often, but it still appears once in a while.
Solid freeform fabrication (SFF) / freeform fabrication (FFF) - The academic community likes these very much. However, the words don't sing and are too much of a mouthful to lead to great popularity. Nevertheless, they're often used in academic papers and patents, and aren't going to go away soon.
Solid imaging / solid imager - 3D Systems and some Japanese companies have used these terms, often applying them specifically to stereolithography, but sometimes meaning the entire field. Solid imaging seems to have fallen by the wayside in recent years, but is still seen once in a while.
Fabber / fabbing - Author Marshall Burns has promoted this term. When he was in academia, we observed his students freely engaging in its use. It leaves open the possibility of an X-acto knife or a milling machine being called a fabber, which they are. In addition, the term is far too redolent of the decade of the sixties. Using the term "fabber" will lead to wanton interpersonal acts, distortions of reality and very loud music.
Layered manufacturing / layered fabrication - All of the commercial systems do indeed operate by making layers and bonding them - some thicker, some thinner. But additive fabrication systems need not do so by definition. There are other ways to add materials together to make things that have been described in the literature. For example, one could build an object in a cylindrical fashion. It's not likely that other such geometries will be of much commercial interest soon, so layered fabrication is not so bad. The term seems mostly to come up in trade magazines when writers are explaining the field to an unfamiliar audience.