The Greek of the Thucydides passage in question (5.116.4) runs οἱ δὲ ἀπέκτειναν Μηλίων ὅσους ἡβῶντας ἔλαβον - "[the Athenians] killed all the hebôntas Melians they captured." The verb hebaô, from which hebôntas is derived, expresses a state that is personified in Hebe, the goddess of youth/adolescence. It is not a strict indication of age, but rather a general description of those who had passed through puberty and reached early adulthood. Both translations you cited are plausible ways to interpret the term. Another way to read it would be "those in their prime".

When the Greeks seized a city, we are often told that they massacred all adult men, but in practice this was just about synonymous with "men of military age." The very oldest, who were no longer of military age, were not particularly worth killing. The purpose of the slaughter, after all, was to exact revenge on that portion of the population that was capable of fighting, and to eradicate an enemy community by destroying its army. The main target of these atrocities was therefore the population bracket that was liable for service in the militia - those between the ages of 18 and 60.

In Thucydides, hebôntas could refer to all men in this age group, or to the younger among them, who were more likely to present a military threat. He uses the verb elsewhere to indicate men in their prime; Athens itself was known to keep the older of its drafted men as a home guard, and use only those below 40 or 45 on missions abroad. If Thucydides were referring to a similar group among the Melians, the correct translation should be "the men in their prime." However, the passage goes on to specify the fate of the women and children captured by the Athenians, and there is no word about the fate of the older men. This is probably why translators have tended to go with a more inclusive term in English, such as "men of military age" (which excludes only a small group of very old men) or "adult men" (which theoretically includes everyone, although the younger among them will have been the priority targets).