Can the "rate coefficient" of a chemical reaction really be called a "rate constant"?
Yes the rate coefficient can be referred to as a "rate constant".
This is in fact a very, very common name for the rate coefficient.  Encyclopedia Brittanica has an entry for rate constant.  Wikipedia has a reaction rate constant page.  Google's N-gram viewer shows that "rate constant" is used more often in the English written corpus by a factor of more than five, relative to "rate coefficient".
The "constant" in rate constant refers to the fact that it does not vary with concentration of the reactants or products.  However, as you note, it could depend on other variables, such as temperature, pressure, catalyst, light, etc.