The paper aims to expand the stability analysis framework based on copositive Lyapunov functions (CLFs) for arbitrarily switching positive systems with continuous- or discrete-time dynamics. The first part focuses on max-type CLFs. It provides an algebraic characterization (in terms of weak quasi-linear inequalities), as well as an existence criterion relying on a concrete construction procedure (in terms of Perron-Frobenius eigenstructure). The second part explores the connections between max-type and two other classes of CLFs, namely linear and quadratic-diagonal. New qualitative and quantitative features are revealed for the CLFs belonging to the two mentioned classes. (C) 2014 Elsevier Ltd. All rights reserved.