The question "is there a subset that sums up to $t$?" has a YES/NO answer. In general, we shouldn't expect an algorithm to do more than it is asked to, so to speak. However, it is rather common that when an algorithm answers YES, it can also naturally give you a certificate that proves it.

Both algorithms you mention explicitly step through certain subsets, and so it is easy for them to also output the subset whose elements sum up to $t$.

The question "is there a subset that sums up to $t$?" has a YES/NO answer. In general, we shouldn't expect an algorithm to do more than it is asked to, so to speak. However, it is rather common that when an algorithm answers YES, it can also naturally give you a certificate that proves it. Moreover, when this happens, you typically get a one valid solution. (This was just a clarification because your example outputs 2 valid solutions. Listing all solutions is a different thing).

Both algorithms you mention explicitly step through certain subsets, and so it is easy for them to also output the subset whose elements sum up to $t$. Whether you care about that extra information is up to you then.