Long wavelength spectral distortions in the Cosmic Microwave Background arising from the 21-cm transition in neutral Hydrogen are a key probe of Cosmic Dawn and the Epoch of Reionization. These features may reveal the nature of the first stars and ultra-faint galaxies that transformed the spin temperature and ionization state of the primordial gas. SARAS~2 is a spectral radiometer purposely designed for precision measurement of these monopole or all-sky global 21-cm spectral distortions. We use 63~hr night time observing of the radio background in the frequency band 110-200~MHz with the radiometer deployed at the Timbaktu Collective in Southern India to derive likelihoods for plausible redshifted 21-cm signals predicted by theoretical models. First light with SARAS 2 disfavors the class of models that feature weak X-ray heating (with $fX\le 0.1$) and rapid reionization (with peak $dTbdz\ge 120 mK\; per\; unit\; redshift\; interval$ ).