In one case Google is retailing the ads and talking to purchasers. The content is subject to Google's review and approval. And, Google has all of the data (quite fine-grained, since that's all they do). Because there is a third party in the transaction (NYT, consumer, Google), it gets the name.

In the latter case, NYT does not share its customer data with google. NYT ad dept is responsible for the editorial and aesthetic signoff of all advertisements. And, arguably, less data collection overall because advertisement is not NYT's business strength.