"Giclee" is an awful term we ought to try to put out of its misery. It was pure marketing BS.
I think most art uses names that describe the medium in a useful manner. The imaging substance and substrate seem to be the most important variables. If either variable is weak, the print, presumably, has less value.
In this respect, "silver gelatin" seems very appropriate, although I think one must distinguish between RC and fiber based paper. I recently saw a high end photo gallery selling "gelatin silver" prints that were obviously on RC paper. I think this is wrong.
With respect to inkjets, the problem has been that there is a huge range of qualities of inks as well as papers. The dyes were very fast fading. The best pigments are very good. From this perspective, I think "pigment" print is more useful than "inkjet" print to describe, for example, an Ultrachrome print.
Although if good pigments are printed on what Epson once called "Archival Matte," the high acidity of the paper should be disclosed, as it will yellow quickly. Most understand the need for "acid free" materials. However, in addition to this, whether a print contains optical brightening agents is also important. These are dyes that will fade quickly and cause the print to yellow.
As a B&W printer, I've been advocating "carbon on cotton" as the best combination -- 100% carbon pigments on an acid-free (and non-brightened) cotton substrate (paper). Carbon is way more lightfast and stable than color pigments.