Given this appears in Gale and Shapley 1962, I'd say it is part of the original formulation (though they do it by saying people are simply not allowed to apply

Given this appears in Gale and Shapley 1962, I'd say it is part of the original formulation (though they do it by saying people are simply not allowed to apply to colleges where they'd be a hard no rather than apply and be rejected). They're looking at stable matchings for college admissions, not perfect matchings.