The problem I see with the simple "Pigment Ink Print" catagorization is that it leaves the door open to non-archival media. While an archival medium is generally presumed, wouldn't "Archival Pigment Ink Print" be more proper if it is indeed archival - which is what "Giclee" and "Fine Art" have traditionally implied?
Using just the lame "Inkjet Print", which specifies only the delivery system, sans content, is equivalent to designating a conventional oil on canvas, masonite, or whatever as "Brushes and Palette Knife" - or "Cans", I guess, for a Jackson Pollock. Not to mention it's immediate association with the ubiquitous desktop AIO.
Pete