Electron energization by magnetic reconnection has historically been studied in the Lagrangian guiding-center framework. Insights from such studies include that Fermi acceleration in magnetic islands can accelerate electrons to high energies. An alternative Eulerian fluid formulation of electron energization was recently used to study electron energization during magnetic reconnection in the absence of magnetic islands. Here, we use particle-in-cell simulations to compare the Eulerian and Lagrangian models of electron energization in a setup where reconnection leads to magnetic island formation. We find the largest energization at the edges of magnetic islands. There, energization related to the diamagnetic drift dominates in the Eulerian model, while the Fermi related term dominates in the Lagrangian model. The models predict significantly different energization rates locally. A better agreement is found after integrating over the simulation domain. We show that strong magnetic curvature can break the magnetic moment conservation assumed by the Lagrangian model, leading to erroneous results. The Eulerian fluid model is a complete fluid description and accurately models bulk energization. However, local measurements of its constituent energization terms need not reflect locations where plasma is heated or accelerated. The Lagrangian guiding center model can accurately describe the energization of particles, but it cannot describe the evolution of the fluid energy. We conclude that while both models can be valid, they describe two fundamentally different quantities, and care should be taken when choosing which model to use.