For the case where the recorded temperature at the DSC plot is taken from the reference pan, rather than from the sample pan, the temperature of initial departure (where the peak starts) should be typically closer to the actual transition temperature. However, quite often, that temperature may be less accurately determined than that of the peak. Consequently, the peak temperature may still be favoured, in practice, to improve repeatability (precision), since that can be better appreciated than accuracy for some purposes. Instead, if the recorded temperature is taken from the sample pan, the transition temperature should be more accurately identified with the peak temperature. However, for the last case, due to its characteristic peak distortion tendency, the repeatability may be somewhat demoted.     

You may possibly want to check the following RG discussion, which also deals with the thermal analysis of common salt: https://www.researchgate.net/post/Thermal_behavior_of_NaCl

I have also read that the melting temperature of a sample (I believe, this is the same as the equilibrium temperature of melting) is found from the Illers diagram, because if the substance is not pure or the peak is very broad than both onset T and peak maximum will be misleading.  Is this correct?

The extrapolation procedure proposed by K.-H.Illers can be checked at the following links:

http://www.sciencedirect.com/science/article/pii/0014305774900275

http://www.masontechnology.ie/x/Usercom_23.pdf

I think that the validity of this procedure is generally arguable; namely, due to the lack of a differential equation-based treatment for the effect of thermal resistance, because it is physically unrealistic to extrapolate the temperature data to a zero mass of sample, and also because it seems hard to estimate the extrapolation error to the isothermal state without leaving the procedure reasoning framework. Nonetheless, its simplicity still seems appealing, while it seems fair to recognize that it has been empirically validated to some extent, particularly for polymer samples.

The onset temperature is indeed less precisely determined, but it is less dependent on the scan rate than the peak temperature.

As said at my first post to this query, if the temperature recorded at the DSC's trace is taken from the sample pan, peak distortion can typically occur. The signalled discussion may possibly be found of some help while dealing with this issue:  https://www.researchgate.net/post/Why_do_I_get_some_endothermic_peaks_denaturation_or_melting_going_backwards_in_temperature_with_the_DSC

Thank you very much for confirmation of my thoughts. I greatly appreciate your input. I believe this is the actual problem I am experiencing here. 

Well defined melting temperature of mixtures can be registered only if you have to do with an euthectic or with a mixture of composition corresponding to an extreme point on the phase diagram.