Cyprus is an outlier among European countries in that the president is both the head of state and the head of government. Currently there are no other countries in Europe with such a system. Even Belarus, with mega-strongman Lukshenko, is still nominally a semi-presidential system.
Why did Cyprus take this kind of constitutional path back in 1960? What influenced this decision?
There is no one answer, it was based on many reasons
Shortly after the founding of the republic, serious differences arose between the two communities of the island(Turkish and Greeks) about the implementation and interpretation of the constitution. The Greek Cypriots argued that the complex mechanisms introduced to protect Turkish Cypriot interests were obstacles to efficient government. In November 1963, President Makarios advanced a series of constitutional amendments designed to eliminate some of these special provisions.
The confrontation prompted widespread intercommunal fighting in December 1963, after which Turkish Cypriots ceased to participate in the government.
The 1960 Cypriot constitution provided for a presidential system of government with independent executive, legislative, and judicial branches, as well as a complex system of checks and balances, including a weighted power-sharing ratio designed to protect the interests of the Turkish Cypriots. The executive, for example, was headed by a Greek Cypriot president and a Turkish Cypriot vice president, elected by their respective communities for 5-year terms, and each possessing a right of veto over certain types of legislation and executive decisions. The Greek Cypriot-controlled Republic of Cyprus retains most elements of the presidential system of government expressed in the constitution, although it has cited the Turkish Cypriots' "withdrawal from government" and the "law of necessity" to enact structural changes that allow "effective governance."
The sate departement have also many info on the politics of Cyprus that can help you.
First of all, Cyprus did not choose. The constitution of Cyprus was provided by the three countries that guarantee Cyprus' existence: the United Kingdom (of which is a former colony), Greece and Turkey (as the population of the island is mostly Greek and Turkish in nationality).
Cyprus was supposed to have power-sharing between Greeks and Turks, based on population. For every 7 Greek civil servants, you get 3 Turkish ones and so on. On that provision, you also needed to have two separately elected leaders with "proportional power to the population of their ethnic group", hence a president and a vice-president. After 1963 the constitution broke down and Turkish Cypriots withdraw to form their own country, the internationally-unrecognised Turkish Republic of Northern Cyprus, leaving the position of vice-president vacant.
Cyprus can also not amend their constitution without the agreement of the three guarantor forces, so any political changes would require solution of the Cypriot Problem first.