Remember how people used to throw the word "thecodont" around to describe some generalized archosaurian ancestor of dinosaurs? The problem with the term was that it had no formal definition--it was just a loose amalgamation of archosaur taxa which had vaguely dinosaurian bauplans. The term "Ornithosuchians" was used for a little while until it was decided that ornithosuchids actually fell to the crurotarsian side of the archosaur family tree. Eventually, thanks to cladistics and common freaking sense, the word "thecodont" went out of style and was replaced by the much more accurate and exclusive name, "ornithodiran." "Ornithodira" has a formal diagnosis. You can tell if something is an ornithodir or not, but that was not the case with "thecodont." This is one of the reasons I love cladistics--it has really put the taxonomy of extinct animals into the limelight. Sure, there are times when you have an animal like Silesaurus or Dracorex where cladistics actually impedes forward momentum, but overall it's quite the system.
Anyway, there's another outdated term floating around the discourse right now: "Mammal-like reptile." The term is used to loosely describe the critters that aren't snakes, lizards, crocodiles, or Petrolacosaurus that eventually gave rise to mammals. Dimetrodon grandis (above) is an oft-cited "mammal-like reptile" that was, in truth, neither a mammal nor a reptile.
"Reptilia" is a formal clade name that includes anapsids and diapsids. The group is defined by features of the skull, mainly, and exactly how many holes are in said skull, and how those holes are arranged. After a long evolutionary history, anapsids are only now represented by turtles, which are themselves a very ancient group of anapsids. Diapsids include lizards, snakes, tuataras, crocodiles, plesiosaurs, non-avian dinosaurs, pterosaurs, Arizonasaurus, birds, and a bunch of other groups. The point is that both the small anapsid group and the huge diapsid group are reptiles. "Mammal-like reptiles" have nothing to do with reptiles.
You see, Timmy, when tetrapods conquered the land and abandoned the water, they became amniotes. Amniotes then split into two main factions--the reptiles, which focused on adaptability, and the synapsids, which focused on...um...their teeth. Both groups layed leathery eggs, and both groups had scaly skin. So why wouldn't they both be called reptiles? Because true reptiles modified their skulls in one way, and synapsids modified their skulls in a different way. So we can diagnose a fossil animal as either synapsid or reptile by looking at its skull. It's an ancient division, and it's the whole reason those labels exist in the first place.
So when people say "mammal-like reptile," what they really mean is "lower-tier synapsid." I'm not an expert on synapsid taxonomy, but I know that "reptile" never shows up in the formal clade definitions. Think about it this way: the word "synapsid" is the equivalent of saying "reptile," but on the mammal side of things. Mammals are synapsids, just like archosaurs are reptiles. It's not hard.
Now stop saying "mammal-like reptile."