Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted subgraph set is the same but the multi set is different?
Harary conjectured that the set of vertex deleted subgraphs is unique up to isomorphism. (On the reconstruction of a graph from a collection of subgraphs. In Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Publ. House Czechoslovak Acad. Sci., Prague, 1964, pp. 47–52.)
If you could find an example, you would have proven Haray's strong reconstruction conjecture false. This is because if the vertex reconstruction conjecture is true, then the edge reconstruction conjecture is true, thus the same would hold for the set versions.
The two problems you mention are not the same, though it is obvious that one implies the other. Nobody knows if they are equivalent. Probably the reason that the "set" version is less known is that it seems to be harder than the "multiset" version version that has already proved too difficult for the many people who have worked on it.