Reduced models are derived for a strongly magnetized collisionless plasma at scales large relatively to the electron thermal gyroradius, in two asymptotic regimes. One corresponds to cold ions and the other to far sub-ion scales. By including the electron pressure dynamics , these models improve the Hall reduced MHD and the kinetic Alfvén wave model of Boldyrev et al. (2013), respectively. We show that the two models can be obtained either within a gyrofluid formalism (Brizard 1992) or as suitable weakly nonlinear limits of the FLR-Landau fluid of Sulem & Passot (2015) which extends anisotropic Hall-magnetohydrodynamics by retaining low-frequency kinetic effects. It is noticeable that, at the far sub-ion scales, the simplifications originating from the gyroaveraging operators in the gyrofluid formalism and leading to subdominant ion velocity and temperature fluctuations, correspond, at the level of the FLR-Landau fluid, to cancellation between hydrodynamic contributions and ion finite Larmor radius corrections. Energy conservation properties of the models are discussed and an explicit example of closure relation leading to a model with a Hamiltonian structure is provided.