The long-time self-diffusion coefficients D(S)(L) in distinctly different glass-forming liquids are analyzed from a unified point of view recently proposed by the present author. It is shown that as long as the systems are in equilibrium, they are all described by the following two types of master curves, depending on whether the control parameter is intensive or extensive: D(S)(L)(x) = d(0)x(-1) (1 - x)(2+eta)exp[cx(3+eta)(1 - x)(2+eta)] for a reduced intensive control parameter x, such as a reduced inverse temperature, and D(S)(L)(x) = d(0)x(-1)(1 - x)(2) for a reduced extensive control parameter x, such as a reduced volume fraction, where d(0) and c are constant. Here, the exponent eta (not equal 0) results from many-body correlations in a supercooled liquid state. The constants eta and c depend on the systems and are given by (eta,c) = (4/3,62) for fragile liquids, (5/3, 62) for strong liquids, and (0,0) for other glass-forming systems in which the peak heights of their non-Gaussian parameters are always much less than 1.0. It is also shown that all of the data for the diffusion coefficient start to deviate from the master curves at lower temperatures (or higher volume fraction), where the systems become out of equilibrium, leading to a glass state.