Rationalising an apparent inconsistency in the determination of reaction spontaneity
I recommend the following article by Juan Quílez if you have access: DOI: 10.1021/ed100477x. The main point is summarized in their figure:



The lower curve shows your first approach, with $\Delta_r G$ as the slope of the curve. The upper curve (i.e. straight line) shows your second approach, where the slope is non-zero and constant. What the second approach leaves out (and it might be appropriate for certain cases, but usually not) is the entropy of mixing (the $R T \ln(Q)$ term in calculating Gibbs energy for given concentrations).

For some more interpretation of the graph and the different definitions, see Orthocresol's excellent answer here: https://chemistry.stackexchange.com/a/41864/72973