 This study investigates the existence and stability of mixed and demixed modes in binary atomic Bose-Einstein condensates with repulsive interactions in a ring-trap geometry, using eigenvalue spectra from Bogolia-Bofter-Genz equations and coupled Gross-Batewski simulations to verify stability. In the one-dimensional limit, analytical analysis reveals stable composite single-peak structures and double-triple-peak ones above a certain particle number threshold. While in the two-dimensional annular geometry, stable radially demixed states can carry arbitrary vorticity and increase stability with increasing vorticity. The study also considers hidden vorticity states in both geometries. This article was authored by Jaoping Chen, Yongyao Li, Nikolaos P. Prukakis and others.