That is the same definition of ternary associativity that I have seen. Such ternary operations are discussed in the papers [1],[2]. Furthermore, the paper [3] surveys some results on n-ary groups.

Ternary associativity can be used to create binary associative operations. This is illustrated in the following proposition. Take note that all the listed identities in the following proposition are some forms of associativity.

$\mathbf{Proposition}$ Suppose that $t$ is a ternary operation on a
set $X$. Then define operations $*,\#$ on $X^{2}$ by letting
$(a,b)*(c,d)=(t(a,b,c),d)$ and $(a,b)\#(c,d)=(a,t(b,c,d))$. Then

- $(X^{2},*,\#)$ always satisfies the identity

$$((a,b)*(c,d))\#(e,f)=(a,b)*((c,d)\#(e,f)).$$

- The following are equivalent.

i. $t$ satisfies the identity $$t(t(a,b,c),d,e)=t(a,b,t(c,d,e)).$$

ii. $*$ is associative.

iii. $\#$ is associative.

- The following are equivalent.

i. $t$ satisfies the identity $t(t(a,b,c),d,e)=t(a,t(b,c,d),e)$.

ii. $(X^{2},*,\#)$ satisfies the identity
$$((a,b)\#(c,d))*(e,f)=((a,b)*(c,d))*(e,f).$$

iii. $(X^{2},*,\#)$ satisfies the identity

$$(a,b)\#((c,d)*(e,f))=((a,b)\#(c,d))\#(e,f).$$

- The following are equivalent.

i. $t$ satisfies the identity $t(a,t(b,c,d),e)=t(a,b,t(c,d,e))$

ii. $(X^{2},*,\#)$ satisfies the identity

$$(a,b)*((c,d)*(e,f))=((a,b)\#(c,d))*(e,f).$$

iii. $(X^{2},*,\#)$ satisfies the identity

$$(a,b)\#((c,d)*(e,f))=(a,b)\#((c,d)\#(e,f)).$$

I proved the above proposition just now to answer this question, so I am unsure if reducing ternary associative operations to binary associative operations has been investigated before (let me know if you find any reference where reducing ternary associative operations to binary associative operations has been studied before). It seems like the above proposition generalizes to n-ary operations as well (I have not verified this yet). Unfortunately, the above proposition does not seem to work very well for having two sided identities for either of the operations $*$ or $\#$. The following proposition (with a trivial proof) relates ternary identities to binary identities.

$\mathbf{Proposition}$ Let $t$ be a ternary operation on a set $X$,
and define binary operations $*,\#$ on $X^{2}$ as before by
letting $(a,b)*(c,d)=(t(a,b,c),d),(a,b)\#(c,d)=(a,t(b,c,d))$. Let
$e_{1},e_{2}\in X$. Then

$(e_{1},e_{2})*(a,b)=(a,b)$ for each $a,b\in X$ if and only if $t(e_{1},e_{2},a)=a$.

$(a,b)\#(e_{1},e_{2})=(a,b)$ if and only if $t(b,e_{1},e_{2})=b$.

[1] Ataguema, H.; Makhlouf, A., Deformations of ternary algebras, Journal of Generalized Lie Theory and Applications, vol. 1, (2007), 41–45.

[2] Ataguema, H.; Makhlouf, A., Notes on cohomologies of ternary algebras of associative type, Journal of Generalized Lie Theory and Applications, vol. 3, no. 3 (2009) 154–174.

[3] Remarks to Glazek's results on n-ary groups
http://arxiv.org/pdf/0704.2749v1.pdf