During geomagnetic disturbances, the solar wind arrives in the form of characteristic sequences lasting from tens of hours to days. The most important magnetic storm drivers are the coronal mass ejections (CMEs) and the slow-fast stream interaction regions (SIRs). Previous data-based magnetic field models did not distinguish between these types of the solar wind driving. In the present work we retained the basic structure of the Tsyganenko and Andreeva (2015) model but fitted it to data samples corresponding to (1) SIR-driven storms, (2) CME-driven storms preceded with a shock ahead of the CME, and (3) CME-driven storms without such shocks. The storm time dynamics of the model current systems has been represented using the parametrization method developed by Tsyganenko and Sitnov (2005), based on dynamical variables W _i, calculated from concurrent solar wind characteristics and their previous history. The database included observations of THEMIS, Polar, Cluster, Geotail, and Van Allen Probes missions during 155 storms in 1997–2016. The model current systems drastically differ from each other with respect to decay rate and total current magnitudes. During SIR-induced storms, all current systems saturate, while during CME-induced disturbances, the saturation occurs only for the symmetric ring current and the tail current. The partial ring current parameters are drastically different between SIR- and CME-induced storm sets. In the case of SIR-driven storms, the total partial ring current is comparable with symmetric ring current, whereas for all CME-induced events it is nearly twice higher. The results are compared with GOES 15 magnetometer observations.