Building upon this question in Math.SE, I think the following might be rather of interest for MO.

In the literature on measure theory, probability and functional analysis the definition of a subset $A \subseteq X$ of a topological space $X$ to be relatively sequentially compact is not unique:

1. $A$ is relatively sequentially compact in $X$ if its closure $\overline{A}$ in $X$ is sequentially compact, i.e. every sequence in $\overline{A}$ has a convergent subsequence (with limit in $\overline{A}$)
2. $A$ is relatively sequentially compact in $X$ if every sequence in $A$ has a convergent subsequence with limit in $\overline{A}$.

In definition 2 it is not really necessary to explicitely demand the limit to be contained in $\overline{A}$ (the limit of any sequence in $A$ is always contained in $\overline{A}$ and even in the sequential closure of $A$).

Clearly, 1 $\Rightarrow$ 2 but for general topological spaces $X$ the converse does not hold as shown in the linked Math.SE thread.

Remark: It seems to be not only the case that a few authors prefer some definition to the other but rather that roughly speaking half of the literature (I was reading so far) uses definition 1 and the other half definition 2. Both have advantages and disadvantages.

Question: Can the spaces $X$ in which 1 $\Leftrightarrow$ 2 for any subset $A \subseteq X$ be characterized by already familiar topological properties?

Note that for the weak topology on a Banach space $X$ these two definitions seem to be equivalent since I found versions of the Eberlein-Smulian theorem which equate weak relative compactness to weak relative sequential compactness with both definitions. But which property of the weak topology on $X$ is really used to equate these two definitions?