We characterize some of the distinctive hallmarks of magnon-magnon interaction mediated by the intracavity field of a microwave cavity, along with their testable ramifications. In general, we foreground two widely dissimilar parameter domains that bring forth the contrasting possibilities of level splitting and level crossing. The former is observed in the regime of strong magnon-photon couplings, particularly when the three modes bear comparable relaxation rates. This character is marked by the appearance of three distinguishable and nonconverging polariton branches in the spectral response to a cavity drive. However, when the bare modes are resonant and the couplings are perfectly symmetrical, one of the spectral peaks gets wiped out. This anomalous extinction of polaritonic response can be traced down to the existence of a conspicuous dark mode alongside two frequency-shifted bright modes. In an alternate parameter regime, where the magnon modes are weakly coupled to the cavity, features of level attraction unfold, subject to a large relaxation rate for the cavity mode. Concurrently, for antisymmetric detunings to the magnon modes, a transmission window springs into existence, exhibiting transparency in the limit of negligible dissipation from the magnons. The emergence of level attraction can be reconciled with a theoretical model that embodies the dynamics of the magnon-magnon subsystem when the cavity field decays rapidly into its steady state. In this limit, we identify a purely dissipative coupling between the magnon modes.