Can someone please give a clear definition and distinction of these terms, as when a public figure is asked a difficult question and says: "I'll take that on notice" or "I'll take that under advisement."
Let me answer your second question first
It's usually used by a judge when he announces he is going to consider what counsel has said and rule on it later.
As for "I'll take that on notice," I haven't heard that used. Usually the term is to "put someone on notice" meaning to alert someone to a fact and to the fact that you have given them some kind of ultimatum:
It's a way of telling someone "You have been warned."
"As for "I'll take that on notice," I haven't heard that used" Just today I have heard it used by a seasoned American civil engineer and constructor as he warned a supervisor of poor crew management. So please feel free.