The von Neumann ordinals are defined in such a way that each ordinal is exactly the set of all smaller ordinals. I am wondering about the origin/motivation for this definition of ordinals (that is, how one got to this definition from the goal of choosing a representative for each equivalence class of well-orderings). I read that the motivation was the fact that each well-ordering is isomorphic to the set of all smaller well-orderings. But when I looked for a proof of this fact, I saw that this proof contained ordinals as a tool to prove it. Now this seemed circular to me (not in the logical sense, but in the historical sense).
Is there also an ordinal-free proof of the fact that each well-ordering is isomorphic to the set of all smaller well-orderings?
Also, I wonder: The definition "An ordinal is the set of all smaller ordinals" would be somehow circular. But would it work rigorously? (Maybe it's some kind of recursive/inductive definition -- these things also seem "circular" but are ok -- also, for example, hereditary sets are defined as sets whose elements are hereditary sets, and this definition also works rigorously.)
Furthermore: How did one get from the slogan "an ordinal is the set of all smaller ordinals" to the definition that an ordinal is a transitive set that is a well-ordering under $\in$?