Bundle up tonight - it's cold out there

"Reptiles" are what's known as a paraphyletic grouping - a group of some but not all of the species descended from a common ancestor. So traditionally, we have thought of reptiles as a group made up of the Lepidosaurs (snakes and lizards) + Anapsids (turtles and tortises) + some of the Archasaurs - ie, the crocodiles and dinosaurs, but not the birds, who are also Archasaurs!

The reason we didn't lump birds in with the rest? Part of it is that there are significant structural differences, but the big reason is that reptiles have usually been described, as you pointed out, as ectothermic ("cold-blooded"), ie unable to use internal means to maintain their metabolisms - and we know that birds are endothermic ("warm-blooded").

Well, more and more paleontologists have been pointing out that dinosaurs were pretty structurally different too, and that there is accumulating evidence that like birds and other warm-blooded animals, some dinosaurs were quite active at high altitudes and in areas with significant yearly snowfall - not what you would expect of "cold-blooded" creatures.

So let T. Rex have his scarf. Besides, it makes him look quite dapper.