The energy gap for electronic excitations is one of the most important characteristics of the superconducting state, as it directly reflects the pairing of electrons. In the copper–oxide high-temperature superconductors (HTSCs), a strongly anisotropic energy gap, which vanishes along high-symmetry directions, is a clear manifestation of the d -wave symmetry of the pairing. There is, however, a dramatic change in the form of the gap anisotropy with reduced carrier concentration (underdoping). Although the vanishing of the gap along the diagonal to the square Cu–O bond directions is robust, the doping dependence of the large gap along the Cu–O directions suggests that its origin might be different from pairing. It is thus tempting to associate the large gap with a second-order parameter distinct from superconductivity. We use angle-resolved photoemission spectroscopy to show that the two-gap behavior and the destruction of well-defined electronic excitations are not universal features of HTSCs, and depend sensitively on how the underdoped materials are prepared. Depending on cation substitution, underdoped samples either show two-gap behavior or not. In contrast, many other characteristics of HTSCs, such as the dome-like dependence of on doping, long-lived excitations along the diagonals to the Cu–O bonds, and an energy gap at the Brillouin zone boundary that decreases monotonically with doping while persisting above (the pseudogap), are present in all samples, irrespective of whether they exhibit two-gap behavior or not. Our results imply that universal aspects of high- superconductivity are relatively insensitive to differences in the electronic states along the Cu–O bond directions.