This paper is the second part of a series of two. It follows [44], in which the positivity-preservation property of methods solving one-dimensional Lagrangian gas dynamics equations, from first-order to high-orders of accuracy, was addressed. This article aims at extending this analysis to the two-dimensional case. This study is performed on a general first-order cell-centered finite volume formulation based on polygonal meshes defined either by straight line edges, conical edges, or any high-order curvilinear edges. Such formulation covers the numerical methods introduced in [6, 32, 5, 41, 43]. This positivity study is then extended to high-orders of accuracy. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. It important to point out that even if this paper is concerned with purely Lagrangian schemes, the theory developed is of fundamental importance for any methods relying on a purely Lagrangian step, as ALE methods or non-direct Euler schemes.