Is it right to consider the tensor product of modules $V$,$W$ and $Z$ to be the vector space of finite linear combinations of $v \otimes w \otimes z$. Thus all elements of the form $\sum_{0}^{N} a_{i,j,k} ( v_{i} \otimes w_{j} \otimes z_{k})$ for any elemets $v,w,z$ of the respective spaces, where we impose the conditions that
$ \alpha (v_{i} \otimes w_{j} \otimes z_{k})= \alpha v_{i} \otimes w_{j} \otimes z_{k}= v_{i} \otimes \alpha w_{j} \otimes z_{k}= v_{i} \otimes w_{j} \otimes \alpha z_{k} $ and
$(v_{i}+v_{l}) \otimes w_{j} \otimes z_{k}=(v_{i} \otimes w_{j} \otimes z_{k})+(v_{l} \otimes w_{j} \otimes z_{k})$
$v_{i} \otimes( w_{j}+w_{l}) \otimes z_{k}=(v_{i} \otimes w_{j} \otimes z_{k})+(v_{i} \otimes w_{l} \otimes z_{k})$
$v_{i} \otimes w_{j} \otimes (z_{k}+z_{l})=(v_{i} \otimes w_{j} \otimes z_{k})+(v_{i} \otimes w_{j} \otimes z_{l})$
in above such sums.
Any post I seen on this goes of the charts by introducing heavy machinery while I think one should be able to think of it as this.
That is correct (for $\alpha$ an element of the base ring).
However, using that as a definition hides the most important property of the tensor product, namely the universal property that it satisfies.