In addition to being listed separately in publications like the CIA World Factbook, Vatican City also has its own ISO-3166 domain: .va. The title for 3166 is "Codes for the representation of names of countries and their subdivisions", which could imply a subdivision of Italy, except...
The United Nations was a party to Vatican City receiving its own domain, as the UN is represented among the voting members of the ISO 3166 Maintenance Agency, and the UN list of country names lists Vatican City as separate and distinct from any other nation, with no comment indicating it is a territory of, or administered by, another nation; that is the case with Montserrat, Faroe Islands, French Guiana, and numerous others.
The website for Stato della Città del Vaticano, translated as Vatican City State, says:
Vatican City State was founded following the signing of the Lateran Pacts between the Holy See and Italy on February 11th 1929. These were ratified on June 7th 1929. Its nature as a sovereign State distinct from the Holy See is universally recognized under international law.
Article 2 of the Lateran Pact states that
Italy recognizes the sovereignty of the Holy See in the international realm as an attribute inherent in its nature in conformity with its tradition and with the requirements of its mission to the world.
Other nations are landlocked, having no natural borders such as an ocean. Vatican City is simply a nation set within the borders of another nation. You should definitely list Vatican City as a separate nation on your itinerary.