I'm calculating the minimum sample size to conduct a One-way ANOVA test that I will follow with a Tukey-HSD post hoc analysis. I'm using software R to do my calculations.

From the 'pwr' package, I need to give as input the number of groups (k), the effect size (f) the significance level, and the power to get the minimum sample size. However, to get the effect size, one needs a pilot study or a guess. From a small pilot study let's say I got the following trial data:

However, how should I compute the effect size from the pilot study? I've seen several "effect sizes": Cohen d, Eta-Squared, Partial Eta Squared, Omega Squared, Epsilon Squared, etc.. I also read in the following book: Practical Statistical Power Analysis using WebPower and R, that they have an online calculator: https://webpower.psychstat.org/models/means03/effectsize.php which gave me the following:

It is also said in the book "Cohen defined the size of effect as: small 0.1, medium 0.25, and large 0.4", and here I got >1, which is really strange. I don't understand why are there so many different effect sizes as output and what is being used for the "overall effect size". Why are there interaction effect sizes, i.e, the effect size for group $x$ vs group $y$? I thought we only analyze the difference between each pair of groups in a Tukey-HSD test.

For example, if the effect size was f=1.0096, I would get the following:

> pwr.anova.test(k =4 , f =1.0096 , sig.level =0.05 , power =0.80)

 Balanced one-way analysis of variance power calculation 

          k = 4
          n = 3.806971
          f = 1.0096
  sig.level = 0.05
      power = 0.8

  NOTE: n is number in each group

This would give around 4 for each group, i.e, 4*4=16 in total. The sample is so small because the effect size of 1.0096 is really big. This leads me to another question: is the effect size of 1.0096 correct?

And for a two-way ANOVA, is the effect size "formula" the same?