Vesicles under shear flow exhibit various dynamics: tank-treading (tt), tumbling (tb) and vacillating-breathing (vb). The vb mode consists in a motion where the long axis of the vesicle oscillates about the flow direction, while the shape undergoes a breathing dynamics. We extend here the original theory [C. Misbah, Phys. Rev. Lett. 96, 028104 (2006)] to the next order in a consistent manner. The consistent higher order theory reveals a direct bifurcation from tt to tb if Ca=tau gdot is small enough-typically below 0.5 (tau= vesicle relaxation time towards equilibrium shape, gdot=shear rate). At larger Ca the tb is preceded by the vb mode. For Ca >> 1 we recover the leading order original calculation, where the vb mode coexists with tb. The consistent calculation reveals several quantitative discrepancies with recent works, and points to new features. We briefly analyse rheology and find that the effective viscosity exhibits a minimum at tt-tb and tt-vb bifurcation points. At small Ca the minimum corresponds to a cusp singularity, while at high enough Ca the cusp is smeared out.