These are great, thanks! These are precisely what I was asking for. However, what do you mean from "different angles"? Each pair is identical, right? Mind you, I think it's more or less clear without the need from another angle, but I spent several minutes looking for the 7 differences.

@VilleSalo I've heard more than once that Følner sets are usually strange, but this is of course personal. I do agree with you on the lamplighter, and it fits the question perfectly.

@YCor you are indeed right that "weird" is subjective. I just meant Folner sequences that are not balls. Hence your example of $\mathbb{Z}$ fits the question, but I'd like some other examples.

Just to add something to the already well written answer, there's an account of the relation between Szemeredi's theorem and Furstenberg-Zimmer's theorem in the excellent book of Kerr and Li about ergodic theory, where they not only treat $\mathbb{Z}$, but general discrete groups acting on locally compact Hausdorff spaces.

@Carl-Fredik, would you have the reference at hand? Or, maybe, just a sketch of the proof.

Thank you to both. @Ycor, what do you mean by "strongly distorted"? Would you have a reference where I could read about Bergman's Property?