Chemical desalination
The method you mentioned is known as directional solvent extraction and seems quite straight forward and efficient. It is a relatively new method of desalination. Soybean oil, hexanoic acid, decanoic acid, and octanoic acid can be used (the last two are most effective) [ source: https://dspace.mit.edu/handle/1721.1/78539 ]. This video shows the setup used.