We provide a unified description, both at the effective and fundamental Lagrangian level, of models of composite Higgs dynamics where the Higgs itself can emerge, depending on the way the electroweak symmetry is embedded, either as a pseudo-Goldstone boson or as a massive excitation of the condensate. We show that, in general, these states mix with repercussions on the electroweak physics and phenomenology. Our results will help clarify the main differences, similarities, benefits and shortcomings of the different ways one can naturally realize a composite nature of the electroweak sector of the Standard Model. We will analyze the minimal underlying realization in terms of fundamental strongly coupled gauge theories supporting the flavor symmetry breaking pattern SU(4)/Sp(4) ~ SO(6)/SO(5). The most minimal fundamental description consists of an SU(2) gauge theory with two Dirac fermions transforming according to the fundamental representation of the gauge group. This minimal choice enables us to use recent first principle lattice results to make the first predictions for the massive spectrum for models of composite (Goldstone) Higgs dynamics. These results are of the utmost relevance to guide searches of new physics at the Large Hadron Collider.