The EOS's explained the EOS page are expressed using KT0 and K0'. The reason for the choice of these two parameters is that these two parameters can be estimated by elastic wave velocity measurement. Although sometimes a higher order EOS, for example, 4th-order Birch-Murnaghan EOS is referred, it is impractical, because the 2nd P derivative of KT is extremely difficult to determine.
Which EOS is the most useful? The 3rd order Birch-Murnaghan EOS is exclusively used, and therefore it is expected that this EOS should provide the most realistic P - V relations based on KT0 and K0'. It is often considered that the Murnaghan's EOS cannot be used for large compression.
Is it true? Let us examine which EOS is the most useful using MgSiO3 bridgmanite as an example, because it is the most important geophysical material and most extensively studied.
Siogeikin et al.  obtained KS0 = 253(3) GPa and Jackson et al.  obtained KS0' = 3.7(3). by Brillouin scattering. Here we assume KT0 = KS0. We depict V/V0 - P relations based on Murnaghan, 2nd-order Birch-Murnaghan, 3rd-order Birch-Murnaghan and Vinet EOSs with these KT0 and KT0' in the figure below. The V/V0 - P relations at ambient T have been studied by Wang et al. , Funamori et al., , Fiquet et al. , Katsura et al.,  and Tange et al., , which are shown as open circles in the figure below.
One may consider that the 3rd-order Birch-Murnaghan EOS should give the best agreement with the V measurement, because it has been extensively used. On the other hand, one may consider that the Murnaghan EOS should give the worst results, because it based on the very primitive assumption. Nevertheless, these expectations are against the examination shown in the above figure. All EOSs give more or less satisfactory results. The Murnaghan EOS shows the best agreement, whereas the 3rd-order Birch-Murnaghan and Vinet EOSs show worse agreements. In my personal opinion, therefore, there is no reason to exclusively use the 3rd-order Birch-Murnaghan EOS, in order to argue the Earth's structure.
What is clear is, on the other hand, that an EOS without considering KT0' is unacceptable.
The main argument of EOS was made more than half century ago. At that time, pressure ranges of laboratory experiments were very limited (at best a couple of GPa). In those days, the Earth's structure should be estimated based on such experimental data obtained at such pressures. The 2nd-order Birch-Murnaghan EOS was some kind of miracle, because it can estimate density of minerals if one has a value of bulk modulus at ambient pressure. These days, the DAC can reach the pressure at the Earth's center, and density of Earth's constituents can be obtained directly. Therefore, argument about EOS has become much less important.