But the quote included suggests the NYT does do the latter: "Half of readers will see one headline, and the other half will see an alternative headline, for about half an hour."
So given that the author mostly observes long consistent blocks of time with the same headline, that suggests the NYT is allocating them to a subgroup in a persistent way (by IP or whatever). Then perhaps the cases where they didn't observe A/B testing were just cases where they were randomized into the optimal (hence final) headline subgroup by accident at the beginning, and never saw any different.