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Moiré patterns of transition metal dichalcogenide (TMD) heterobilayers have proven to be an ideal platform to host unusual correlated electronic phases, emerging magnetism, and correlated exciton physics. While the existence of novel moiré excitonic states is established through optical measurements, the microscopic nature of these states is still poorly understood, often relying on empirically fit models. Here, combining large-scale first-principles GW-BSE calculations and micro-reflection spectroscopy, we identify the nature of the exciton resonances in WSe$_2$/WS$_2$ moiré superlattices, discovering a surprisingly rich set of moiré excitons that cannot be even qualitatively captured by prevailing continuum models. Our calculations reveal moiré excitons with distinct characters, including modulated Wannier excitons and previously unindentified intralayer charge-transfer excitons. Signatures of these distinct excitonic characters are confirmed experimentally via the unique carrier-density and magnetic-field dependences of different moiré exciton resonances. Our study highlights the highly non-trivial exciton states that can emerge in TMD moiré superlattices, and suggests novel ways of tuning many-body physics in moiré systems by engineering excited-states with specific spatial characters.