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Two-color emitting colloidal semiconductor nanocrystals (NCs) are of interest for applications in multimodal imaging, sensing, lighting, and integrated photonics. Dual color emission from core- and shell-related optical transitions has been recently obtained using so-called dot-in-bulk (DiB) CdSe/CdS NCs comprising a quantum-confined CdSe core embedded into an ultrathick (similar to 7-9 nm) CdS shell. The physical mechanism underlying this behavior is still under debate. While a large shell volume appears to be a necessary condition for dual emission, comparison between various types of thick-shell CdSe/CdS NCs indicates a critical role of the interface "sharpness" and the presence of potential barriers. To elucidate the effect of the interface morphology on the dual emission, we perform side-by-side studies of CdSe/CdS DiB-NCs with nominally identical core and shell dimensions but different structural properties of the core/shell interface arising from the crystal structure of the starting CdSe cores (zincblende vs wurtzite). While both structures exhibit dual emission under comparable pump intensities, NCs with a zincblende core show a faster growth of shell luminescence with excitation fluence and a more readily realized regime of amplified spontaneous emission (ASE) even under "slow" nanosecond excitation. These distinctions can be linked to the structure of the core/shell interface: NCs grown from the zincblende cores contain a similar to 3.5 nm thick zincblende CdS interlayer, which separates the core from the wurtzite CdS shell and creates a potential barrier for photoexcited shell holes inhibiting their relaxation into the core. This helps maintain a higher population of shell states and simplifies the realization of dual emission and ASE involving shell-based optical transitions.