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In Discovering Statistics Using R by Andy Field (2012) it seems that the effect size is $\eta$:

For some bizarre reason, in the context of ANOVA, $r^2$ is usually called eta squared, $\eta^2$. It is then a simple matter to take the square root of this value to give us the effect size, r ($\sqrt{.46} = .68$) (p.454)

From other source, e.g. Wikipedia, it seems that the effect size is $\eta^2$.

Because there are thresholds to separate small from medium from large effects it makes a difference whether $\eta$ or $\eta^2$ "measures" the effect. In the example from the book cited above you have $\eta^2$ = 0.46 (medium effect) and $\eta$ = 0.68 (large effect).

Which of the two variables actually measures the effect size?

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An effect size is the measure of difference between groups or the measure of the effect of a variable.

Using this definition, both variables can be considered to be an effect size. It is important, however, to specify which one you use when reporting the effect size.

I have no source for that, simply common sense.

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