“Pagan antiquity was the age of status; Christian mediaevalism was the age of vows; and skeptical modernity has been the age of contracts; or rather has tried to be, and has failed” (“The Story of the Vow,” in The Superstition of Divorce). He goes on to illustrate these distinctions by highlighting the difference b/w an ancient slave, a Christian vassal, and a modern contract-laborer.
I would bet, though it’s just a bet at this point, that Chesterton is consciously emending the two-stage modernization theory offered by Henry Maine, which Fukuyama discusses in The Origins of Political Order. For Maine, modernization is essentially a transition from status-based to vow-based societies; Chesterton injects a distinctively Christian third mode of social organization, around vows.
Interestingly, other Christian genealogists of modernity — Milbank and MacIntyre, for instance — also seem to like three-stage histories, from pagan antiquity, to a Christian synthesis, to a de-Christianization (which might also be a return to paganism in some respects) in modernity. Non-Christian genealogists of modernity (Weber and Marx, for instance), understandably less inclined to highlight Christian exceptionalism, tend to have two-stage histories in which modernity emerges from a relatively undifferentiated past.