A precising definition is a definition that contracts or reduces the scope of the lexical definition of a term for a specific purpose by including additional criteria that narrow down the set of things meeting the definition.
For example, a dictionary may define the term "student" as "1. anyone attending an educational institution of any type, or 2. anyone who studies something."
However, a movie theater may propose a precising definition for the word "student" of "any person under the age of 18 enrolled in a local school" in order to determine who is eligible to receive discounted tickets.
This type of definition is useful in preventing disputes that arise from the involved parties using different definitions of the term in question.
A precising definition is intended to make a vague word more precise so that the word's meaning is not left to the interpretation of the reader or listener.