“The Vatican” is commonly used as a name for the Holy See, the central governing body of the Catholic Church and Vatican City. The Holy See encompasses the office of the pope and serves as the Church’s spiritual and administrative authority.
In the news report, “the Vatican” refers to the Church authority issuing updates about Pope Leo’s recovery and planned engagements. The Holy See is distinct from Vatican City, the sovereign city-state in Rome where it is headquartered.