A first inversion chord must have the third chord factor in the bass, but it may have any arrangement of the root and fifth above that, including doubled notes, compound intervals, and omission (E-G-C, E-G-C-G', E-C'-G'', etc.)
A first inversion chord must have its third note in the bass, but it can be built with any combination of root and fifth above that, including doubled notes, compound intervals, and omissions.