The term Kerr black hole refers to a particular black hole model that includes the effects of its mass and rotation but not charge, and the term is likely sometimes used for actual rotating black holes that have significant rotation and little electric charge. The model was developed in the 1960s, decades subsequent to the development of models dealing with mass and mass plus charge. It is generally assumed that charged black holes rapidly lose their charge given their additional attraction of the opposite charge, and practically speaking, the Kerr black hole concept is a practical model for actual black holes. The Kerr metric is the metric devised for the model, and Kerr spacetime refers to the spacetime model it defines.
Rotation is a significant complication to general relativity (GR) based models, introducing frame dragging, the ergosphere, and revealing the potential for the Penrose process and Blandford-Znajek mechanism and affecting other characteristics such as the photon sphere and innermost stable circular orbit.
The Kerr-Newman black hole is an extension to the Kerr black hole model that accommodates electric charge, also developed in the 1960s. The earlier (1920s) models that accommodated only mass and only mass with charge where the Schwarzschild black hole and the Reissner-Nordström black hole.
While the Kerr black hole is based upon GR, it can be valid for some theoretical modifications to GR, so determining that an actual black hole shows characteristics of a Kerr black hole does not necessarily distinguish GR from some amended version of GR.