Before we proceed with your question, I'd like to give an interesting example to clarify a point. We say that a set $Y$ is cofinite in a superset $X$ of $Y$ if and only if $X\smallsetminus Y$ is finite. Observe that a union of cofinite sets is cofinite by Demorgan's laws (since an intersection of finite sets is finite), and that a finite intersection of cofinite sets is likewise cofinite (since a finite union of finite sets is finite). Consequently, we can describe a cofinite topology on any set $X$, by saying that $Y\subseteq X$ is "open" if and only if $Y$ is cofinite or $Y$ is empty. Here's where it gets interesting: If $X$ is an infinite set with the cofinite topology, then a sequence of points converges to every point of $X$ if and only if it has infinitely many distinct terms--for example, considering $\Bbb R$ in the cofinite topology, the sequences $1,\frac12,\frac13,\frac14,...$ and $1,0,\frac12,0,\frac13,0,...$ converge to every point of $\Bbb R$!!
In these (and other) problematic spaces, we don't have uniqueness of sequence limits, so saying $\lim\limits_{n\to\infty}x_n$ is not meaningful in general, even if it is known that $\{x_n\}$ converges. To avoid this, it is sufficient (but not necessary) that a space be Hausdorff.
Let's say (to distinguish between the two definitions) that $x^*$ is a cluster point of $\{x_n\}$ if it is a limit point of some subsequence of $\{x_n\}$. Note that I said a limit point--the discussion above demonstrates why that's relevant. This generalizes your proposed alternate definition for "accumulation point" to general spaces.
Now suppose $x^*$ is a cluster point of the sequence $\{x_n\}$. Take any open set $U$ containing $x^*$. Since $x^*$ is a cluster point of $\{x_n\}$, then there is some $\{x_{n_k}\}$ such that $x_{n_k}\to x^*$. Then for sufficiently large $k$, each $x_{n_k}\in U$ by definition of sequence convergence, which implies that there exist infinitely many $k$ (so infinitely many $n_k$) for which $x_{n_k}\in U,$ and so there exist infinitely-many $n$ for which $x_n\in U$. Thus, every cluster point is an accumulation point.
As pointed out above, the converse need not hold in general, so they are related, but not generally equivalent concepts. Again, as pointed out above, first-countability is sufficient for the equivalence of the two concepts (and may be necessary, too, but I'm not sure about that).