The spherical Cauchy distribution and the Poisson-kernel based distribution were both proposed in 2020, for the analysis of directional data. The paper explores both of them under various frameworks. Alternative parametrizations that offer numerical and estimation advantages, including a straightforward Newton-Raphson algorithm to estimate the parameters are suggested, which further facilitate a more straightforward formulation under the regression setting. A two-sample location test, based on the log-likelihood ratio test is suggested, completing with discriminant analysis. The two distributions are put to the test-bed for all aforementioned cases, through simulation studies and via real data examples comparing and illustrating their performance.