Off-axis final states of cylindrically bounded 2D fluids can develop from initially unstable, but cylindrically symmetric, 2D vorticity distributions. Experiments with electrons in a Malmberg-Penning trap, as well as 2D fluid simulations, demonstrated that such states result when the initial vorticity distribution is close to the boundary, while less extended distributions lead to on-axis states. A simple thermodynamic model, maximizing the entropy of a state consisting of a diffuse background surrounding a strong coherent vortex, is shown to quantitatively predict this bifurcation, while conserving circulation, angular momentum, and energy.