The present-day expansion rate of the universe as constrained by Type Ia supernovae (SNe Ia) is directly tied to the zero point of the extragalactic distance scale, which is in turn set by the calibration of astrophysical distance indicators such as the Tip of the Red Giant Branch (TRGB). In this study, a new calibration of the TRGB luminosity is determined in the Magellanic Clouds (MCs). Composite TRGB measurements are constructed for the Small and Large MCs from individual regions in which the TRGB could be unambiguously identified. As a result, the highest accuracy measurement of the TRGB in the Clouds to date is determined. The I vs. (V-I) relation of the TRGB is found to be consistent with a constant I magnitude over colors 1.45 < (V-I) < 1.95 mag, and a shallow, quadratic curvature is confirmed when including more metal-rich (up to (V-I) = 2.2 mag) Tip stars into the fit. The new measurements are calibrated from apparent brightness to absolute luminosity with reddening maps from the OGLE-IV survey based on observations of the Red Clump (RC), and the Araucaria Project's latest high-accuracy geometric distances based on detached eclipsing binaries (DEBs). This study's TRGB measurements place novel constraints on the three-dimensional tilt of the LMC as well as the distance between the Small and Large Clouds. Both findings are in agreement with the independent, geometric constraints derived from the DEBs and establish a better than 0.02 mag cross-consistency (1% in distance) between the latest DEB measurements, RC reddening maps, and the TRGB measurements of this study.