A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. In this new paper (arXiv:1410.7336), after reviewing the classification of finite-dimensional real Lie algebras of Hamiltonian vector fields on the plane, we present new Lie-Hamilton systems with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the hereafter called planar diffusion Riccati systems and complex Bernoulli equations, all of them with t-dependent real coefficients. Furthermore, we study the existence of local diffeomorphisms among new and already known Lie-Hamilton systems on the plane. In particular, we show that the Cayley-Klein Riccati equations describe as particular cases well-known coupled Riccati equations, second-order Kummer-Schwarz equations, Milne-Pinney equations, the harmonic oscillator with t-dependent frequency and other systems of physical and mathematical relevance.