On the matter of what definition is preferable. Let me add that the same question may be asked for the countably version, that has  *$\omega$-accumulation point* in place of *limit of a subsequence*:  


3. $A$ is relatively countably compact in $X$ if its closure $\overline{A}$ in $X$ is countably compact, i.e. every sequence in $\overline{A}$ has a $\omega$-accumulation point (in $\overline{A}$).

vs

4. $A$ is relatively sequentially compact in $X$ if every sequence in $A$ has a $\omega$-accumulation point (in $\overline{A}$).

I think 2 and 4 are more standard than the variants 1, resp. 2; in fact it seems to me there are a number of reasons to prefer them. 

 * *Property of language*. Definitions (2) and (4) really describe relative properties, whereas (1) and (3) are just cases of the notion of sequential, resp. countable compactness, referred to the space $\overline A$. 

* *Economy of language.* Why squandering locutions that can be used for situations (2) and (4), while (1) and (3) can be simply referred to as  “$\overline A$ is sequentially/countably compact” ?

* *Topological invariance.* Properties (2) and (4) seem somehow more relevant because they behave better under continuous maps. If $A\subset X$ is (2) resp (4) in $X$, and $f:X\to Y$ is a continuous map, then $f(A)$ is (2) resp (4) in $Y$. The same is not true for the odd counterparts (1) and (3).

* *Prevalence of the situation.* Here I'm vague, but I think situations (2) and (4) are quite common and relevant in the mathematical practice, because dealing with sequences taken from the set $A$ vs sequences taken from $\overline A$ can make a difference. The closure of $A$ in a weak topology may contain unknown wild objects (think of a subset $A$ of a Banach space $E$ and its closure in $E^{**}$ in the $\sigma(E^{**},E^*)$ topology, for instance). 
  
* *Relevance in connection with important theorems*. It only comes to my mind the Eberlein-Shmulian theorem, but I think this great theorem alone, a Northwest Passage of Functional Analysis, is enough to ask for a special term for situations (2) and (4).