Let $A_\bullet$ be a dg-algebra over a field $k$. Let $M_\bullet$ (resp. $N_\bullet$) be a right (resp. left) $A_\bullet$-module. There is then a notion of the **derived tensor product**:
$$M_\bullet\otimes^L_{A_\bullet}N_\bullet=\bigoplus_{p\geq 0}M_\bullet\otimes(A_\bullet)^{\otimes p}\otimes N_\bullet$$
(where $\otimes$ is $\otimes_k$) with differential given by the internal differential on each direct summand plus an alternating sum of all possible maps multiplying adjacent elements (decreasing $p$ by one). We can rewrite this as:
$$(M_\bullet\otimes^L_{A_\bullet}N_\bullet)_j=\bigoplus_{p\geq 0}(M_\bullet\otimes(A_\bullet)^{\otimes p}\otimes N_\bullet)_{j-p}$$
But now (based on my limited understanding) it is not clear to me why we shouldn't instead take direct product instead of direct sum, and define:
$$(M_\bullet\otimes^L_{A_\bullet}N_\bullet)_j=\prod_{p\geq 0}(M_\bullet\otimes(A_\bullet)^{\otimes p}\otimes N_\bullet)_{j-p}$$
Note that with this definition the differential still makes sense.

The definition with the direct sum seems to be standard. On the other hand, I have encountered a context where I want to write down a certain element in $M_\bullet\otimes^L_{A_\bullet}N_\bullet$, but it lies in the direct product (and not in the direct sum). Hence my question is:

Which of the two definitions of $M_\bullet\otimes^L_{A_\bullet}N_\bullet$ given above is correct, and why?

Note that when $A_\bullet$ is concentrated in nonnegative degrees, taking direct product instead of direct sum makes no difference, since for fixed $j$, the factors on the right become trivial for sufficiently large $p$. Hence this question is only non-vacuous when $A_\bullet$ is nontrivial in negative degrees.