DFA for the set of strings of 0’s and 1’s such that there are two 0’s separated by a number of positions that is a multiple of 4
Your interpretation is correct. It looks to me like the "solution" is wrong (and the solution isn't strictly a DFA). 

Assuming you don't mind an NFA (getting a DFA is a bit fiddly, but obviously doable), you can fix the solution by removing the $(q_{4},0) \rightarrow q_{5}$ transition, and adding $(q_{5}, 0) \rightarrow q_{5}$ and $(q_{5}, 1) \rightarrow q_{5}$ transitions.

After some fiddling, it seems like a minimal DFA for this has 23 states... not so much fun.