Step 1: Understand the question.
We are asked who gave the well known definition of the State as the body that holds the monopoly of legitimate force over a territory.
Step 2: Recall the source.
This line comes from the lecture 'Politics as a Vocation' by the German sociologist Max Weber. He defined the State by its means, which is the claim to legitimate use of physical force, and not by its aims.
Step 3: Check option 1, Karl Marx.
Marx saw the State as a tool of the ruling class that protects its economic interests. This is a class view, not the monopoly of legitimate force. So option 1 is wrong.
Step 4: Check option 2, Max Weber.
Weber's definition matches the quoted words closely. So option 2 is right.
Step 5: Check option 3, Emile Durkheim.
Durkheim wrote about social facts, collective conscience and the division of labour. He saw the State as an organ of moral thinking for society, not as the holder of force. So option 3 is wrong.
Step 6: Check option 4, B. R. Ambedkar.
Ambedkar was a jurist and social reformer who drafted the Constitution. He did not give this definition of the State. So option 4 is wrong.
Final Answer:
The definition belongs to Max Weber, which is option 2.
\[ \boxed{\text{Option 2: Max Weber}} \]