What does it mean to modify or test Newton's second law?

Physicists in the 19th century concluded that Newton's second law is not a real law of nature but a mere definition, because force cannot be measured directly but only through the acceleration it produces on a given mass. Although the debate seemed settled, it has been proposed in recent years that Newton's second law could be modified, as if it were a real description of nature, to explain some intriguing aspects of galactic dynamics. Experiments have been performed claiming to test the validity of the second law. The aim of this article is to elucidate, without using the concept of force, what it means to modify or experimentally test Newtonian dynamics.