Difference between standard and extended Steinhart-Hart equation
Actually the general form of the Steinhart-Hart equation is an infinite sum (see German Wikipedia article of Steinhart-Hart equation):

\$\frac{1}{T} = \sum_{k=0}^{\infty} c_k \ln^k \frac{R}{R_{ref}} \$

In most practical cases it is enough to approximate by using only 3 of the first 4 coefficients \$c_0, c_1, c_3\$ (in your case called \$A, B, C\$) and throw away the rest (as \$c_2\$ often is small enough to neglect and the terms with higher powers of \$\ln \frac{R}{R_{ref}}\$ become less and less significant).

In your case, obviously, in the "extended" equation just some extra effort was taken to also use coefficient \$c_2\$ in order to get a better approximation.

Probably the coefficients are obtained not only by measuing at a 4th temperature but by doing many (>4) measurements at many temperatures in the interesting range and doing least-square curve fitting using a 4th order polynomial as model function.