Volcano deformation monitoring is fundamental to detect pressurizations of magma bodies and forecasting any ensuing eruptions. Analytical and quasi-analytical solutions for pressurized cavities are routinely used to constrain volcano deformation sources through inversion of surface displacement data. Due to their computational efficiency, such solutions enable a thorough exploration of the parameter space and thereby provide insight into the physics of magma-rock interaction. Developing more general deformation models can help us better characterize subsurface magma storage. We develop quasi-analytical solutions for the surface deformation field due to the pressurization of a finite (triaxial) ellipsoidal cavity in a half-space. The solution is in the form of a non-uniform distribution of triaxial point sources within the cavity. The point sources have the same aspect ratio, determined by the cavity shape, while their strengths and spacing are determined in an adaptive manner, such that the net point-source potency per unit volume is uniform. We validate and compare our solution with analytical and numerical solutions. We provide computationally-efficient MATLAB codes tailored for source inversions. This solution opens the possibility of exploring the geometry of shallow magma chambers for potential deviations from axial symmetry.