Many of the standard abstract mathematical structures were first defined and studied "externally" (in terms of some sort of concrete representation) and only later defined "internally" (as abstract spaces obeying some list of axioms), after a foundational theorem had been established demonstrating the equivalence of the two definitions. For instance (oversimplifying the history a little bit to emphasise the point):

1. Groups were first defined as permutation groups, or as groups that could be represented by permutations. It was only with Cayley's theorem that one could equivalently define such groups in terms of the modern group axioms.
2. In the early study of manifolds (e.g., by Poincaré), these spaces were often understood to be subspaces of some ambient Euclidean space, as opposed to the modern internal definition using atlases of coordinate charts, etc.. With the advent of embedding theorems such as the Whitney embedding theorem (for smooth manifolds) or Nash embedding theorem (for Riemannian manifolds), one could relate the two types of definitions. In a similar spirit, Gauss's theorema egregium equates the external and internal definitions of what we now call Gauss curvature.
3. Lie algebras were initially studied as algebras of linear transformations, often finite dimensional in nature. The modern abstract definition came later, with theorems such as Ado's theorem and its relatives (Poincaré-Birkhoff-Witt theorem, Engel's theorem, Lie's theorem, etc.) providing fundamental equivalences between the two viewpoints. I believe the history of von Neumann algebras follows a similar trajectory, though I am less familiar with this story.
4. Boolean algebras are an interesting case in that (from my understanding of the history), Boole introduced the abstract concept of this algebra first, before the realisation that concrete Boolean algebras of sets obeyed Boole's axioms. Nowadays of course, due to Stone's theorem, the two definitions can be viewed as equivalent.

Many of the standard abstract mathematical structures were first defined and studied "externally" (in terms of some sort of concrete representation) and only later defined "internally" (as abstract spaces obeying some list of axioms), after a foundational theorem had been established demonstrating the equivalence of the two definitions. For instance (oversimplifying the history a little bit to emphasise the point):

1. Groups were first defined as permutation groups, or as groups that could be represented by permutations. It was only with Cayley's theorem that one could equivalently define such groups in terms of the modern group axioms.
2. In the early study of manifolds (e.g., by Poincaré), these spaces were often understood to be subspaces of some ambient Euclidean space, as opposed to the modern internal definition using atlases of coordinate charts, etc.. With the advent of embedding theorems such as the Whitney embedding theorem (for smooth manifolds) or Nash embedding theorem (for Riemannian manifolds), one could relate the two types of definitions. In a similar spirit, Gauss's theorema egregium equates the external and internal definitions of what we now call Gauss curvature.
3. Lie algebras were initially studied as algebras of linear transformations, often finite dimensional in nature. The modern abstract definition came later, with theorems such as Ado's theorem and its relatives (Poincaré-Birkhoff-Witt theorem, Engel's theorem, Lie's theorem, etc.) providing fundamental equivalences between the two viewpoints. I believe the history of von Neumann algebras follows a similar trajectory, though I am less familiar with this story.
4. Boolean algebras are an interesting case in that (from my understanding of the history), Boole introduced the abstract concept of this algebra first, before the realisation that concrete Boolean algebras of sets obeyed Boole's axioms. Nowadays of course, due to Stone's theorem, the two definitions can be viewed as equivalent.
5. Probability spaces are still commonly defined today using a concrete representation (a sample space $\Omega$, equipped with a sigma-algebra of events and a probability measure). But these spaces (up to almost sure equivalence) can be equated also with commutative tracial von Neumann algebras, thanks to the classification theory of the latter. This equivalent definition is the most convenient starting point for introducing noncommutative probability spaces, which do not enjoy a classical representation in terms of sample spaces (though, thanks to the GNS construction, one can still interpret them in terms of algebras of bounded operators on a Hilbert space).