I) The term exterior multiplication ("äussere Multiplication") is due to Grassmann, who introduced the term in his book (written in 1844)

*Die Wissenschaft der extensiven Grösse oder die Ausdehnungslehre, eine neue Mathematische Disciplin"*

As you can check in the table of contents of the book (on page 276), paragraphs §§34,35 are called *Grundgesetze der äusseren Multiplication* (Basic laws of exterior multiplication).
Here is the scan of this book by Google .

II) The terminology exterior differential ("différentielle extérieure") was introduced in the 1930's by articles of Elie Cartan, inspired by Grassmann.

Here is a secondary reference from an Analysis course by Chatterji and another by Chern and Chevalley, in their analysis of Elie Cartan's mathematical contributions (cf. in particular pages 229 and 230 )