Is he who vomited intentionally during the day in Ramadan due to stomach pain—without putting anything in his mouth—considered among those who were overcome by vomiting and is not required to make up the day?

A fasting person breaks their fast if they intentionally vomit, and they must make up for that day, based on the hadith: "Whoever is overcome by vomit, does not have to make it up; but whoever induces vomit, must make it up." However, whoever is overcome by vomit does not break their fast and does not have to make up for that day.