So I learned that after an ANOVA of that kind you want to do a post hoc test, which in this case of 3 levels could be either Tukey or a recoding into dummy variables of 2 of the 3 levels. I am new to this so I only see that Tukey includes the difference between level 2 and three whereas the dummy coding doesnt show that contrast.

I am assuming I need to apply correction for multiple testing to both.. but as you probably see I am very naive in this and dont know which is the best thing to do. I have the feeling that dummy coding has another primary purpose than this. Happy for any insight. Thank you.