What is the definition of a "Simple enzyme" exactly? Is it an enzyme that does not require a co-factor? I'm having a hard time making a distinction between simple and complex enzymes. the definitions I've found so far all say this:


  Simple enzymes are enzymes that are only made out of protein, and complex enzymes are ones that, in addition to the protein portion, have a non-protein portion as well, which is called a co-enzyme/prosthetic group (I've seen both terms being used)


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)?
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.