Linnæus and Laurenti and other 18th century taxonomists, as we now call them, had not much to go on other than morphology and behaviour of existing critters. Fossils were known but the field of systematic palaeontology could not even be said to be in its infancy, and DNA would not even be discovered for another 200 years.
The calls they made were reasonable given what was known at the time and some of the taxa they described are still known and basically unchanged. Some have gone by the wayside as more has been learnt, and through the 20th and 21st centuries there were major reconfigurations of the taxa as the molecular and fossil evidence rolled in. In second half of the 20th century, Huxley and Hennig and others advocated a strict cladistic approach, by which any taxon should be monophylic: that is, any taxon should consist of a crown and all its descendants. In some cases the crown may be something not specifically known from the fossil record but that can be referred to in abstract: eg you could define it as the most recent common ancestor of, say, coelacanths and lungfish. For comparison, paraphylic taxa involve part of the descent from an ancestor but with some excluded groups. Polyphylic groups include parts of multiple chains of descent.
The cladistic approach was an important influence over the past half-century. There are now no polyphylic vertebrate taxa and a scarce few paraphylic taxa.
The one major paraphylic taxon among the vertebrates remaining that still gets a bit of play: a lot of refs will still list the surviving classes of tetrapods as Class Amphibia, Class Mammalia, Class Aves, and Class Reptilia. (Note: Amniota is a clade containing mammals, birds, reptiles.)
But strictly, Reptilia is bunk from a cladistic understanding. Even from morphology, a close look at crocodiles indicates that they are more closely related to birds than they are to turtles, tuatara, snakes, or lizards, and fossil evidence and molecular analysis backs this up to the point where it is not disputed. There is therefore no clade that can contain turtles AND crocodiles, but not birds.
There is also a minor issue related to certain extinct ancestors of the modern mammals: Reptilia includes early therapsids but not their descendants the modern mammals, so that is also a cladistic no-no.
How to deal with this is a matter of dispute and various ideas have come up, among them:
1/ Do nothing, just leave Reptilia as it is for old time’s sake
2/ A phylocode-friendly idea to split off those early “mammal-like reptiles”, and call Reptilia all the descendants of the most recent common ancestor of turtles and crocodiles. So this would include snakes, lizards, tuatara, turtles, crocodiles … AND birds.
3/ Just forget about Reptilia: stop talking about it altogether, it is a classical relic of a term. Amniota would be divided into Synapsida (which includes mammals and the mammal-like “reptiles”), and Sauropsida. Sauropsida is divided into Anapsida (which has turtles and such) and Diapsida. Diapsida is Lepidosauromorpha (including lizards, snakes, tuatara), Archosauromorpha (includes crocodiles and birds) and some other extinct stuff.
The removal of Reptilia from formal literature would not change the fact that “reptiles” would remain a word used in ordinary life. Class Pisces was ditched over a hundred years ago once it was understood not to be a meaningful group from a phylogenic point of view, but of course people still talk about “fish”. For that matter, people use terms referring to all kinds of polyphylic groups: “sea creatures”, “rats”, etc.
FTR, the most recent common ancestor of birds and crocodiles lived about 250 million years ago. The most recent common ancestor of crocodiles and turtles lived about 310 million years ago.