What is the definition of a "Simple enzyme" exactly? Is it an enzyme that does not require a co-factor?
What is the definition of a “Simple enzyme” exactly?


This is not a term that is commonly used. There are commonly used terms for an enzyme lacking its cofactor (apoenzyme) and having its cofactor (holoenzyme). Also, the term "enzyme complex" is used for the enzyme bound (non-covalently) to other molecules.


  Does this mean that any protein that utilizes a co-factor, even if that co-factor is simply a co-enzyme or an activating metal, i.e., not an actual part of the enzyme's structure, is considered a complex enzyme? Or are the two terms restricted to the structure of the enzyme (i.e., whether they have prosthetic groups, be they organic or inorganic, or not)?


Yes, just apply the definition you were given. If the metal/co-enzyme/prosthetic factor is absent, the enzyme does not work.