"Saprophyte" refers to a mode of nutrition. Saprophytes gain their nutrition from formerly living and now dead organisms and their parts. The "-phyte" or "phyto-" roots just mean a plant or plant-like organism. Don't be too strict and narrow here. I don't consider fungal or bacterial decomposers as really being plants, but they have that historical connection.
"Epiphyte" takes that root word but refers to life habit, that is something that grows on a plant and "endophyte" refers to a fungus or bacterium that lives inside a plant.
Our wonderful language, both technical and colloquial, is derived from multiple sources, all with their own traditions in how best to combine roots to form new terms. Consequently, don't expect parallel construction! I might say that with respect to mode of nutrition, that the meadow button mushroom is a saprophyte, that the cedar-apple rust fungus is a parasite (takes its food from another living creature without providing an apparent benefit) as well as an obligate biotroph (it's food source needs to be alive) and a pathogen (causes disease), and that the apple tree is an autotroph (meaning makes its own food). Sure, you make it a little more parallel by calling the button mushroom a "saprogen" or "saprotroph" (with the latter being how I was taught during some of graduate school). Still, it's not always going to line up perfectly with respect to word construction and syntax.
Actually, the history of forest pathology and even all of biology itself hinges precisely on the concepts behind these words, but I reckon I've killed off discussion and any residual interest in this topic has vaporized!