Given definition: Given a sequence, $(a_n)$, then the series with terms $(a_1,a_2,...,)$ is a sequence $(s_n)$ of partial sums.
Does this mean essentially the definition above can be re-stated to say a series $$(s_n)_{n \in \mathbb{N}} = (s_1, s_2, s_3, ..., s_n) = (a_1, a_1 + a_2, a_1 + a_2 + a_3, a_1 +... + a_{n-1} + a_n)$$
Or have I mis-interpreted the definition?
The left hand side has $n\in\Bbb N$, which means an infinite series, while the middle and right hand side are finite series. Which do you mean?
You also have some confusion in variables. In the left hand side, $n$ is the index variable (dummy variable), while in the middle and the right hand side it is the index of the final item in the series. Fixing the finite-versus-infinite problem should also take care of that.
If you fix the left side, then your definition of a finite series seems to be correct. Summation notation would make it clearer, however:
$$s_k=\sum_{i=1}^k a_i$$