The usual interpretation is, as you suggest, wife's family name is second, and, at that period, hyphenated. However, it's entirely possible that family members (the children) will be found without the second part of the name. One must be open to all possibilities!
It is not a question of "adopting" a hyphenated name in any formal sense, by some official act; rather, it's a matter of fashion and social convention. The use of the second part is optional. The fashion is not especially old, I don't recall seeing it with any frequency before 1800.
Also, we frequently find that given names are subject to fashion as well. When the person has more than one given name, in some areas there is a tendency for the person to use the second of those given names in adult life, but we actually find any and all combinations and permutations of the given names in official records, newspapers, etc., and sometimes, the person may use a completely different given name as an adult. Just part of the fun!