We recently proposed a technique able to represent the spatial variations of the magnetic field at regional scales. However, we pointed out that these preliminary developments were not suited for the complete representation of the geomagnetic field. In this paper, we propose a complete revision, the revised spherical cap harmonic analysis (R-SCHA), which introduces slight changes in order to rectify the previous shortcomings. In addition, some discussions shed a new light on the former spherical cap harmonic analysis (SCHA) and help us to demonstrate its deficiencies and approximations. We finally show that R-SCHA now fully satisfies the natural properties of potential fields. R-SCHA also yields analytical relationships with the spherical harmonics. Taking advantage of the mathematical equivalence of both representations, we explore the relevance of fundamental concepts like spectrum, minimum wavelength, or internal/external field separation. We conclude that these concepts are misleading and must be handled with care in regional modeling. A prime goal being the ability of R-SCHA to represent real data sets, we also investigate and illustrate the effect of finite series expansions. A norm for the regularization of the inverse problem is proposed as well. The conclusions drawn in this paper allow us to validate the method and to assert that the present proposal is suited for modeling and studying the lithospheric magnetic field from ground to satellite altitudes at regional scales.