Global structural identifiability determines if it is possible to uniquely estimate unknown parameters of a model from measurements. We consider two definitions for global structural identifiability - one proposed by Walter and Lecourtier in Mathematics and Computers in Simulation, 60: 472-482 (1982) and the other by Glad and Ljung in the Proceedings of the 29 th Conference on Decision and Control (1990). The two definitions appear distinct because of the role of the inputs. We present here a proof of the equivalence of the two definitions. We revisit the formulation of the optimization problem to analyze global structural identifiability proposed by Asprey and Mantalaris in IFAC Computer Application in Biotechnology (2001) and propose a modification to the problem formulation. We also demonstrate with the help of an elementary model that their solution algorithm can lead to erroneous conclusions. We further analyze the global structural identifiability of a simple model using the optimization-based approach. (C) 2019 Elsevier Ltd. All rights reserved.