Answer:  

Lawrence J. Diggs, who is actually studying and teaching about vinegar, kindly answered the question above:
   
   

"Artificial vinegar" is acetic acid that is made by a chemical process. "Natural vinegar " is acetic acid that is made in a biological process using the Acetobacter aceti bacteria. If the "natural vinegar" is distilled, it is very difficult to tell the difference between it and the "artificial vinegar". 
   

Various types of vinegar Courtesy of  Gizmo Highway

 
"Natural vinegar" can be made from just about anything with enough sugar or starch. Many things; like wood, fruits, vegetables, roots, honey, milk, grass, grains and more are used. The law in the United States and many industrialized countries designate that anything called "vinegar" must be made in a natural process from natural materials. 
 
The "artificial vinegar" is called "acetic acid". It is used industrially to make products like pharmaceuticals and plastics. For example acetic acid is used to make aspirin. 

 
Reference
Vinegar: The User Friendly Standard Text, Reference and Guide to Appreciating, Making, and Enjoying Vinegar  
  
 
Acknowledgment
We would like to thank Lawrence J. Diggs for his kind answer.