To be precise and reiterate daniel-r-h's point, these constructions aren't really how the naturals are "defined". The naturals are defined up to isomorphism by whatever number axioms you're using, of which both the simple nesting sequence implied by Shieldfoss's question and the one in the OP are models. The simple nesting is the Zermelo numerals, and the more complex one is the von Neumann construction of the ordinals.
The names spoil why the von Neumann construction is preferred. Both model the naturals, but the von Neumann construction is really a model of the ordinals, of which the naturals are an initial segment. This is because while a Zermelo numeral is a set containing only its predecessor, a von Neumann ordinal is the set containing all of its predecessors. This gives the nice property that in this model < is ∈, and ≤ is ⊆. For example, 4 is the set {0, 1, 2, 3} (shown above), ω is the set ℕ, and ε₀ = ⋃(ω, ω^ω, ω^ω^ω, … ).
In contrast, the Zermelo numerals "stall at the first limit", and cannot be extended in this way. Since a Zermelo numeral contains only its immediate predecessor, it cannot express limit ordinals which have none.
