There is no difference between "ionization energy" and "first ionization energy". Another definition from Wikipedia:
The ionization energy (IE) is qualitatively defined as the amount of energy required to remove the most loosely bound electron, the valence electron, of an isolated gaseous atom to form a cation.
Here it's more clear that they are the same, since "the most loosely bound electron" is a single electron, and each successive ionization only removes a single electron at a time.
Additionally, if one were to remove all electrons from an atom, it would be explicitly stated as "complete ionization".