SUM: Let $S_*'$ and $S_*''$ be subcomplexes of $S_*$. Then $S_*'+S_*''$ is the subcomplex of $S_*$ whose $n$th term is $S_*'+S_*''$.
DIRECT SUM: Let $\{(S_*^{\lambda}, \partial^{\lambda}) : \lambda \in \Gamma\} $ be a family of complexes. Their direct sum is the complex $$... \rightarrow \sum_{\lambda} S_{n+1}^{\lambda} \rightarrow ^{\partial_{n+1}}\sum_{\lambda} S_{n}^{\lambda} \rightarrow ^{\partial_{n}} \sum_{\lambda} S_{n-1}^{\lambda} \rightarrow \dots$$ where $\partial_n = \sum_{\lambda} \partial_n^{\lambda}: \sum_{\lambda}s_n^{\lambda} \mapsto \sum_{\lambda} \partial_n^{\lambda}(s_n^{\lambda})$ for $s_n^{\lambda} \in S_n^{\lambda}$.
I'm having trouble understanding the difference between these two definitions.
For a sum, I see that $S_*' + S_*''$ means the indirect product of the two subgroups of $S_*$, as in $S_*'+S_*''=\{x+y : x \in S_*' \text{ and } y \in S_*''\}$.
But what does the summation notation mean when the groups involved aren't subgroups of a larger group? If $S_n(X)$ and $S_n(Y)$ are members of the family, what does $S_n(X) + S_n(Y)$ mean if they're different groups? And how would you define $x + y$ as a sum of elements from this group?
The notation used in your definition of "direct sum" is quite nonstandard. As you say, normally a notation like $\sum_{\lambda\in I} A^\lambda$ is only meaningful if all the abelian groups $A^\lambda$ are subgroups of a single group $A$, so you can take the sum inside $A$. Here that notation is being used with a different meaning, which is usually written $\bigoplus_{\lambda\in I} A^\lambda$ instead. This denotes the subgroup of the product $\prod_{\lambda\in I} A^\lambda$ consisting of elements with only finitely many nonzero coordinates. We can consider each $A^\lambda$ as a subgroup of $\bigoplus_{\lambda\in I} A^\lambda$, namely the subgroup of elements all of whose coordinates are $0$ except possibly the $\lambda$ coordinate. Note that then $\bigoplus_{\lambda\in I} A^\lambda$ is the internal sum of these subgroups isomorphic to $A^\lambda$ inside the group $\prod_{\lambda\in I}A^\lambda$.