What is the definition of sanity? How can I prove someone either sane or insane? In my experience the definition of insanity is thus:


  Insanity: The state of being seriously mentally ill, mad, and/or irrational.


Is there a proper scientific definition of this term? While mentally ill is relatively easily to pin down, the others criteria seem subjective. 

Is there any test or method that can be used to prove a subject to be either sane or insane? How can any such test give objectively accurate results for this or another similar definition of insanity?
Sanity is an explicit legal definition. It is generally not a psychological term. This is Wikipedia's definition of Sanity which aligns perfectly to my understanding of abnormal psychology. (emphasis mine):


  In criminal and mental health law, sanity is a legal term denoting that an individual is of sound mind and therefore can bear legal responsibility for his or her actions. The official legal term is compos mentis. It is generally defined in terms of the absence of insanity (non compos mentis). It is not a medical term, although the opinions of medical experts are often important in making a legal decision as to whether someone is sane or insane.


Psychologists are often called in to determine "sanity" but they deduce whether one fits the legal definition based on any psychological disorders the individual might have. "Insanity" in this sense could be considered to be one of many psychological disorders that would cause one to be unable to function in a court room and understand their rights.

"Temporary Insanity" was a previously common but largely ill regarded legal excuse in the earlier days of psychology. It is not a psychological term nor does it have a psychological definition.

You'll note the DSM does not include diagnostic criteria for sanity. It is not a term commonly used in research or psychiatry because as you suggest it is much too broad of a concept. 

Research on "insane" people is as uselessly broad as it would be on "silly" or "dumb" people. Instead research is conducted on specific conditions or measures. 

Wikipedia has some information on Sanity in Psychiatry and Psychology but note these are very outdated ideas and I'm not aware of any current researchers who respect these theories as valid.