A form is defined to be alternating iff having two equal arguments means it is equal to 0. It is defined to be anticommutative iff permuting its arguments means multiplying by the sign of the permutation.
Unless my definitions are wrong, I'm pretty sure the two are equivalent, right? Also, if it is the case, why do we use two different names for the same concept?
Expanding upon the comment, alternating always implies anticommutative, and the converse is true provided that $2$ is invertible in your ground ring (e.g. field of characteristic $0$ or $p>2$).
Alternating $\implies$ Anticommutative
Suppose that $\def\o{\omega}\def\e{\eta}\def\w{\wedge}\o \w \o = 0$ for all forms $\o$. Then for any forms $\o, \e$, $$ \o \w \e + \e \w \o = (\o + \e) \w (\o + \e) - \o \w \o - \e \w \e = 0. $$
Anticommutative $\implies$ Alternating provided that $2^{-1}$ exists
Suppose that $\o \w \e + \e \w \o = 0$ for all forms $\o, \e$. Then for any form $\o$, $$ 2(\o \w \o) = \o \w \o + \o \w \o = 0. $$ If $2$ is invertible in the ground ring, then by multiplying by $2^{-1}$, we have $$ \o \w \o = 0. $$
