With Gianluca Paolini (in preparation), we constructed families of strongly minimal Steiner (systemsforeveryk 3.Aquasigroupisastructurewithabinaryoperationsuchthatforeachequationxy=zthevaluesoftwoofthevariablesdeterminesauniquevalueforthethird.Hereweshowthatthe2^{ Steiner (2,3)-systems are definably coordinatized by strongly minimal Steiner quasigroups and the Steiner (2,4)-systems are definably coordinatized by strongly minimal SQS-Skeins. Further the Steiner (2,4)-systems admit Stein quasigroups but depending on the choice of theory may or may not admit a definable binary function and be definably coordinatized by an Steinquasigroup.WeexhibitstronglyminimaluniformSteinertriplesystems(withrespecttotheassociatedgraphsG(a,b)(CameronandWebb)withvaryingnumbersoffinitecycles.Weshowhowtovarythetheorytoobtain2or3$-transitivity. This work inaugurates a program of differentiating the many strongly minimal sets, whose geometries of algebraically closed sets may be (locally) isomorphic to the original Hrushovski example, but with varying properties in the object language. In particular, can one organize these geometries by studying the associated algebra. This work differs from traditional work in the infinite combinatorics of Steiner systems by considering the relationship among different models of the same first order theory.