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Mean-field theory is an approximation replacing an extended system by a few variables. For depinning of elastic manifolds, these are the position of its center of mass $u$, and the statistics of the forces $F(u)$. There are two proposals to model the latter: as a random walk (ABBM model), or as uncorrelated forces at integer $u$ (discretized particle model, DPM). While for many experiments ABBM (in the literature misleadingly equated with mean-field theory) makes quantitatively correct predictions, the microscopic disorder force-force correlations cannot grow linearly, and thus unboundedly as a random walk. Even the effective (renormalized) disorder forces which do so at small distances are bounded at large distances. We propose to model forces as an Ornstein Uhlenbeck process. The latter behaves as a random walk at small scales, and is uncorrelated at large ones. By connecting to results in both limits, we solve the model largely analytically, allowing us to describe in all regimes the distributions of velocity, avalanche size and duration. To establish experimental signatures of this transition, we study the response function, and the correlation function of position $u$, velocity $\dot u$ and forces $F$ under slow driving with velocity $v>0$. While at $v=0$ force or position correlations have a cusp at the origin, this cusp is rounded at a finite driving velocity. We give a detailed analytic analysis for this rounding by velocity, which allows us, given experimental data, to extract the time-scale of the response function, and to reconstruct the force-force correlator at $v=0$. The latter is the central object of the field theory, and as such contains detailed information about the universality class in question. We test our predictions by careful numerical simulations extending over up to ten orders in magnitude.