Hi, I create these and mainly call my sets backgrounds.
Background images don’t have to be seamless, but sometimes they are. They’re decorative images you can use as digital background images (but they can have other applications and can be printed if they’re high enough resolution). I’m more inclined to call them backgrounds if they’re the sort of images you can use as backdrops or put text or other images in front of, but they can be images of their own accord. Some people use the term textures interchangeably with backgrounds and others prefer to reserve it for more textured images.
Digital papers are pretty much the same thing, so you could use the terms interchangeably. When I sell items as digital papers, I usually have more emphasis on them for printing/crafting purposes, but they don’t really differ from backgrounds because I create them all at a print resolution (at least 300PPI/300DPI).
With seamless images, it means you can tile them and the edges meet up, so you can use them for purposes like web site backgrounds or on products for sale (where you’d tile them to fill a larger area than the image itself). Not all seamless images are patterns, and not all patterns are seamless. Usually seamless patterns are created in a way that they can be tiled, and both the pattern itself and the background behind the pattern meets up, so you can tile them horizontally and vertically.
Note: That’s my interpretation so it might not be ‘textbook’.