On page 4 of Many Lives of Lattice Theory the author wrote: "Two equivalence relations on a set are said to be independent when every equivalence class of the first meets every equivalence class of the second." I fail to understand this definition, because in lattice of partitions we define meet and join of sets of equivalence classes, not individual classes.
An equivalence class is a set, so it makes sense to talk about the intersection with other sets.
Yes, intersection of sets of equivalence classes can also be defined, but here they are not used.
If you are actually interested in why this definition is used, you need to give more context.
Edited to add:
At the end of the paragraph cited in the question Rota explains: "Two equivalence relations are independent when the answer to either question gives no in- formation on the possible answer to the other question." (The question is: In which equivalence class of the equivalence relations does my object lie?)
So he gives the definition and the motivation. Your problem seems to be that in other contexts one is interested in other definitions which is true, but so what?