I find that presentations are the most useful tool to get an intuitive feeling about the nature of some group construction. I have read here at mathstack that given $\{G_i = \langle X_i \mid R_i \rangle\}_{i \in I}$ a family of groups with presentations given, we have $$\prod_{i \in I} G_i = \left\langle\left. \bigcup_{i \in I} X_i \phantom{.}\right| R \cup \bigcup_{i \in I} R_i\right\rangle,$$ where $R = \{[x_i, x_j] \mid i, j \in I, x_i \in X_i, x_j \in X_j\}.$ Here, i'm assuming that the unions are disjoint so to prevent any formal complications, but you get the idea.
Using the same family of groups, one can define $$\bigoplus_{i \in I} G_i = \{((g_i)_{i \in I} \mid g_i = 1_{G_i} \text{ for all but finitely many } i\}.$$ This is a subgroup of the product, and I was wondering how one can create a presentation for it using the presentations for each $G_i$. The problem here is how to, using relation, impose the fact that all but finitely many coordinates vanish.
That presentation of the direct product is incorrect if the indexing set $I$ is infinite. There is no easy way to write down a presentation in general if $I$ is infinite.
However, it is a correct presentation of the direct sum, so your question is already answered. In a presentation you are only multiplying finitely many generators at a time so it is already true that you can't get the entire direct product but only get the direct sum.