Distinguishing insulators by the dominant type of interaction is a central problem in condensed matter physics. Basic models include the Bloch-Wilson and the Peierls insulator due to electron-lattice interactions, the Mott and the excitonic insulator caused by electron-electron interactions, and the Anderson insulator arising from electron-impurity interactions. In real materials, however, all the interactions are simultaneously present so that classification is often not straightforward. Here, we show that time- and angle-resolved photoemission spectroscopy can directly measure the melting times of electronic order parameters and thus identify-via systematic temporal discrimination of elementary electronic and structural processes-the dominant interaction. Specifically, we resolve the debates about the nature of two peculiar charge-density-wave states in the family of transition-metal dichalcogenides, and show that Rb intercalated 1T-TaS(2) is a Peierls insulator and that the ultrafast response of 1T-TiSe(2) is highly suggestive of an excitonic insulator.