Definitions of dielectric constant and relative permittivity? How do they differ?

Question

It's all about the definition, as in how they both are defined.

My book defines dielectric constant as the proportionality constant
$$K=\frac{\sigma}{\sigma-\sigma_p}$$
And then defines permittivity as
$$\epsilon=\epsilon_0K$$

Many websites on the Internet including Wikipedia say that they are defined the same. 

On the other hand, this website says that they are distinguished for their scope - the dielectric constant is only defined for dielectrics, but the relative permittivity is defined for all materials.

My question is, how do we define them into the convention used today? And how do you then define permittivity or is it the other way round?