EDIT: This answer was posted in response to the original version of the question.
The first part of your question, and the question in your title, appears to be answered in the Wikipedia article on Francoist Spain that you cited:
"... officially known as the Spanish State (Spanish: Estado Español) from 1936 to 1947 and the Kingdom of Spain (Spanish: Reino de España) from 1947 to 1975"
The Wikipedia article on "Spanish State" redirects to Francoist Spain because that was the official name of the state from 1936 to 1947 (i.e. during the "Francoist" era)
The legal basis for the change from "Spanish State" to "Kingdom of Spain" would appear to be Article 1 of the Ley de Sucesión en la Jefatura del Estado (Law of Succession to the Headship of the State) which states:
"Spain, as a political union, is a Catholic, social and representative State that, according to its tradition, is declared a Kingdom"
So, after that law was passed in 1947, Spain became officially known as "The Kingdom of Spain".
The Wikipedia article on Francoist Spain that you cited has a link to the Law of Succession to the Headship of the State (via the 1947 referendum on that law) in the section titled Government.
The 1947 Law of Succession to the Headship of the State is one of the eight Fundamental Laws of the Realm established in Spain between 1938 and 1977.