Semi-analytical models of disk outflows have successfully described magnetically-driven, self-confined super-Alfv\'enic jets from near Keplerian accretion disks. These Jet Emitting Disks are possible for high levels of disk magnetization $\mu$ defined as $\mu=2/\beta$ where beta is the usual plasma parameter... In near-equipartition JEDs, accretion is supersonic and jets carry away most of the disk angular momentum. However, these solutions prove difficult to compare with cutting edge numerical simulations, for the reason that numerical simulations show wind-like outflows but in the domain of small magnetization. In this work, we present for the first time self-similar isothermal solutions for accretion-ejection structures at small magnetization levels. We elucidate the role of MRI-like structures in the acceleration processes that drive this new class of solutions. The disk magnetization $\mu$ is the main control parameter: massive outflows driven by the pressure of the toroidal magnetic field are obtained up to $\mu \sim 10^{-2}$, while more tenuous centrifugally-driven outflows are obtained at larger $\mu$ values. The generalized parameter space and the astrophysical consequences are discussed. We believe that these new solutions could be a stepping stone in understanding the way astrophysical disks drive either winds or jets. Defining jets as self-confined outflows and winds as uncollimated outflows, we propose a simple analytical criterion based on the initial energy content of the outflow, to discriminate jets from winds. We show that jet solution are achieved at all magnetization level, while winds could be obtained only in weakly magnetized disks that feature heating. (read more)