There needs to be a distinction between "civil rights" (lower case) and "Civil Rights". The former is simply a noun phrase that refers to the natural rights that all citizens should have respected by their government, while the latter is the name of the Civil Rights era, along with the associated laws and arguments surrounding the specific issue of racism in America during the 1950's and beyond. Given America's history of slavery and Jim Crow in the Old South, it's natural for Americans to conflate the terms when they are speaking, since racism is such a prominent feature of modern history and of current politics.
When it comes to civil rights for other groups (women, the LGBT community, etc.), there are those that wish to draw a distinction between the hardships they suffer, and those suffered by blacks throughout most of American history (similar to Holocaust references -- there is little that can compare to the Holocaust). There are others who argue that racism, sexism, homophobia, etc. are all different forms of discrimination that need to be eradicated and want them all recognized under the same umbrella (the author of your linked article takes this position).
But no matter which side of this argument you're on, the fact that there are sides to be taken is proof that the concept has two definitions in America.