Assume there was a fantasy world in which a ruler was elected to become "king". This king election would be completely democratic and fair. Then, when the king is elected, he or she has absolute power in the kingdom and nobody could stop him/her from making decisions.
What would the proper terms be for this kind of government?
You are proposing a combination of two different dimensions. Each dimension has its own terminology.
A country in which the leader is selected by voting is called a democracy. A narrow usage of the word democracy means only that some important decisions are made through voting. In some usages, it may carry different assumptions (for example, Kant's view of democracy includes a strong protection of civil liberties).
A country in which one person or body holds all meaningful political power is called an autocracy. Usually people think of dictatorships or monarchies when they think of autocracies.
Although these two dimensions are often thought to be related, they are not necessarily. Thomas Hobbes in "Leviathan" (one of the most famous books in social contract theory, which underlies much of democratic theory) an entire book to this subject. According to Hobbes, both elected rulers and conquerors have the same rights and restrictions. In both cases, the ruler and ruled did not enter into a contract - and so the ruler cannot be bound by anything (this synopsis lays out his argument in more detail).
There are other examples from political theory also.
For what it is worth, this system of government has been in place in many counties in the U.S. state of Georgia for much of its history. At the local government level, it is known as the sole commissioner system.
Territories formed under the Northwest Ordinance of 1787 in the United States initially had leaders appointed by the President with the advice and consent of the Senate, rather than elected ones, but no legislative branch. Territorial governments were run by a territorial governor, a territorial secretary, and three judges.
There are also numerous historical examples of elected kings.
If the proposed form of government only holds elections upon the death or resignation of the ruler, it is an absolute monarchy or despotism, with an elected monarch.
If the proposed form of government has regularly scheduled elections (such as every year, every two years, every four years, or every six years), it is a republic.
Some countries that have had similar systems: From about 1290 to about 1618, the Holy Roman Empire was an elective monarchy with 7 electors. During Renaissance and early modern times, Poland was an elective monarchy (elected by a large class of nobles). Neither the HRE nor Poland had anything like an absolute monarchy, though. Haiti under the Duvaliers had an elected president for life.
In countries with elective monarchies, a major historical theme is that the ruler seeks to choose his (or her) successor. Often the ruler manages to change the law of succession, either to replace the elective monarchy with a hereditary monarchy, or to restrict the list of eligible candidates to members of his (or her) family. Alternatively, the ruler can designate their successor while they are still in power. This is the purpose of "crown princes", and of Mexico's tradition of the President choosing his party's candidate at the next presidential election.
This form of government can be used to legitimize the results of succession wars that occur after the previous monarch dies. The winner of the war becomes the "winner of the election" instead of a mere usurper. During early modern times, the Ottoman Empire and the Mughal Empire had a tradition of succession wars after emperors died. Bohemia had two major succession wars during the seventeenth and eighteenth centuries; the winners of these wars were legitimized by the fact Bohemia had an elective monarchy.
For examples of how this form of government can be used to transition from a revolutionary republic to an Empire, consider the elections and plebiscites that elected Napoleon Bonaparte and Napoleon III in France.
