Were college sports professional or amateur in the second half of the 19th century? Who gets to define each category?
In the second half of the nineteenth century, college sports, while they were played by amateurs, did not prohibit professionals from entering into its ranks. In their early history, college sports were not perceived nearly as favorably as they are in today's society, and colleges certainly did not go out and attempt to lure students to their schools because of their athletic prowess. To be successful as sports, some colleges would hire professionals to fill their teams. That being said, colleges did not have teams comprised entirely of professional players. When the NCAA was established in the first decade of the twentieth century, the use of professionals in college sports was officially prohibited. Since that time, collegiate sports have been confined to amateurs.
As far as who should define each category, that is a more difficult question. Most would agree that amateur athletes are those who do not play a given sport as a source of income; professionals, on the other hand, do exactly that. Clearly, with the establishment of the NCAA, the presiding body has the power to delineate amateurs and professionals. A separate party (not from among the athletes) must make the decision, to avoid any question of a conflict of interests.