A DE or double edge is a type of safety razor. Much like the gillette sensor's, Mach III's, and Fusion's are also safety razors. They are safety razors in that they usually have some sort of safety function (usually in the form of a safety bar or a fixed angle razor) built in them to prevent the user from slicing their face open. A straight razor, which has no safety function (simply a razor that can be used at any angle on the face) is not considered a safety razor. Instead it is referred to...pardon the expression...a cutthroat. Hope this helps.
I think that it is better to classify a safety razor as a DE or injector--the head is the same only the blade is changed. The AOS subclassifies Mach 3 and such as cartridge razors and since the head and blades as changed at the same time I think it is appropriate but subtle distinction.