Diet coke has no sugar, no fat, and less than 100 mg sodium. The specific results from Nature are about trehalose, a sugar not found in diet coke. I'm not aware of any diet soda that contains it, either. I realize it's an opinion piece, but pseudoscience and speculative articles should not be the domain of NYT.
Of note, the zero-calorie definition of artificial sweeteners is based on the amount excreted in urine/stool: It should be very close to the ingested amount. I find it hard to believe the conclusion drawn by the authors that other sweeteners (sucralose and saccharin), are consumed by bacteria since the amount ingested matches the amount excreted.