Can a consensus be reached on the exact meaning of "Cohen's D" and "Hedge's G"?

Question

This question: Difference between Cohen's d and Hedges' g for effect size metrics shows there are at least two different interpretations each of both Cohen’s D and Hedge’s G, one of them in common (the one with weighted pooled standard deviation calculated using weights proportional to n-1, where n is the sample size). This confusion (at least in the moment when I am writing) may also be seen in this Wikipedia page: https://en.wikipedia.org/wiki/Effect_size , where this common formula is shown for both statistics (in the case of Hegde’s G, a star is added to $s$ in the standard deviation, but then the formulas for $s$ and $s^*$ are the same). It seems to me the situation is even more complicated than what seen in Stack Exhange’s questions. In fact, I see several websites using an unweighted version of $s$ (so that it is simply the squared root of the mean between $s_1^2$ and $s_2^2$): https://www.statisticshowto.com/cohens-d/ ; https://www.socscistatistics.com/effectsize/default3.aspx ; https://www.statology.org/interpret-cohens-d/ ; https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/hedgeg.htm . I see here: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3840331/ that this is defined as “common language effect size” (on the StackExchange search engine, by typing "common language effect size" I find 14 questions on it). Given the confusion arising from these contrasting definitions, wouldn’t it be possible to reach a consensus to solve this issue?

Answer 1

I find this article a good starting point. The author uses subscripts to distinguish the different version of Cohen's d.

https://www.frontiersin.org/articles/10.3389/fpsyg.2013.00863/full