I think the best way to answer this question is to define each of the terms that you listed, and the compare them to term "management". Lets start by defining "management":
"the act of overseeing or directing something"
Now, lets define "process":
"a particular course of action intended to achieve a result"
Clearly, if you are going to direct something - such as a group of people - then you are intending to achieve a result. In order to achieve that resul, you will need a course of action. So, the very idea of management is that it is a process. The two are inextricably intertwined.
Next, we will define "discipline":
"a system of rules of conduct or method of practice" OR "a branch of knowledge"
Both of these definitions lend themselves to what we have already said about management. If you are going to direct something, there should be a system of rules in place so that you are both consistent and effective. It is impossible to learn a sport without having a system of rules to follow - otherwise, it would be impossible to play. There would be no way to know who wins - no one would agree upon what action equals a point. The same is said in business - management is its own branch of knowledge because there is system of rules that has to be enacted, understood by all players, and followed.
A human activity sounds pretty straightforward - any activity that is commonly performed by humans. Management is performed by humans in all areas of life. Management of a household, a business, a school, a sport. Even personal relationships require some form of management to keep things running smoothly. As a species, we are heirarchial by nature, and have incorporated principles of management into our everyday lives.
Finally, a career. A career is defined as:
"the particular occupation for which you are trained"
If we can agree that management is a discipline, something that has a branch of knowledge, then we can agree that it is something a person can be trained to do. Looking at monster.com or at any job posting location, we can find numerous listings that come under the heading "management". Individuals who are skilled at managing and directing many people and tasks simultaneously are well-suited to this career.