Conflicting definitions: What is “effect size” regarding customer churn?

One research paper says an example of "effect size" is the difference in the average age of churners vs. non-churners (41 vs 45). Another research paper says "effect size" is the difference in the area under the ROC curve (AUC was used to measure the validity of a model used to predict churn).

What does "small effect size" mean in the context of customer churn? I understand the basic concept of statistical power, and that the sample size needs to be larger when the effect size is smaller - and that the significance criteria is another factor. Does effect size refer to the result (AUC, etc.), any of the independent variables, or either?

Also, how do you calculate the effect size in terms of Cohen's D, when the study is validated with Area under the ROC?
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