We obtain necessary and sufficient conditions for the strongly Lp well-posedness of three abstract evolution equations, arising from fractional Moore-Gibson-Thompson type equations which have recently appeared in the literature. We use Fourier multiplier techniques to derive new characterizations in terms of the R-boundedness of the operator-valued symbol associated to each abstract model, when endowed with the time-fractional Liouville-Grünwald derivative. As a consequence of our characterization, we give new insights into the differences between the models based on the structure of the respective operator-valued symbols and show novel applications by including several classes of operators other than the Laplacian.