I've been laughing at print descriptions for 30 years. The first time I saw "silver gelatin print" I about fell over laughing, and thought "how pretentious." Then I saw "chromogenic" print for a color C-print, and then "dye destruction print" for an Ilfochrome. I've come to the conclusion that describing the print process may be okay since it follows the tradition of both non-photographic and photographic print making: lithograph, etching, monotype, collotype, palladium print, VanDyke print, gum bichromate print - you get the idea.
If you can accept the descriptions of wet darkroom technology - you'll have to give the digital folks a little slack while they sort out the terminology for a digitally created print.
Giclee is sooo stupid (sorry Graham Nash but it's true) and is really only applicable for a print from an Iris printer since that's what the term was coined for originally. I kind of like "pigment print" for an inkjet print using archival pigment inks.
As for the descriptions you've mentioned - digital C-Print works for me if it's from a LightJet, Lambda, Frontier, etc. - even "digigraph" or maybe "digitype" would be okay.