Definition of reptile

Close, but there is a small problem with your definition, that being that birds are within the reptile group, as they are descendants of dinosaurs, or more precisely, non-avian dinosaurs.

To expand your definition a bit, you should more appropriately say everything more closely related to (pick your favorite modern reptile) than to (pick your favorite mammal).

Another way to describe it would be any descendant of the last common ancestor of turtles, snakes and lizards, and crocodilians. Of course, this would leave out some fossil species that might predate this.

What is often done is say that "Reptilia" includes the last common ancestor of all the modern reptiles, as mentioned above, and "Sauropsida" as everything more closely related to modern reptiles than mammals. So in this case, Reptilia is a subset of Sauropsida.

The problem with including something like "amniotic egg" as part of the definition is that it may not match with the relationships well. For instance, Aves had often deen defined as animals with feathers, but this could not work after we found dinosaurs with feathers. So instead, we try to restrict our definitions to strictly relationships.

Nevertheless, we do DIAGNOSE groups by certain characters, so it would be correct to say that all reptiles have amniotic eggs. This would be a diagnostic character of the group, but it would not DEFINE the group.

There has been a lot of confusion even within the scientific field between definitions (what it is) and diagnoses (how do we tell what it is). There have been many many papers written on this very topic and it is not one that is easily sorted out, but it is critically important for clarifying what we are talking about.