The Zurich-London agreements of 1959 between Greece, Turkey and the United Kingdom as the occupying colonial power were the product of compromise between these actors and less as a product of Greek-Cypriot and Turkish-Cypriot involvement. This is an important premise as they're to be envisaged as an international treaty rather an internal agreement on their form of government.This is why Cyprous was often dubbed as the Reluctant Republic, since the majority of its inhabitants preferred a union with Greece or Turkey rather than independence.

Initially a federal state proposed by Turkey was rejected by Greece as more likely to lead to separation. The presidential system was chosen as a compromise and better suited to secure the guarantees that were requested for both communities by their "mother states". This is because the political system was designed to accommodate collective ethnic communities and not single citizens, as the latter were defined by the former and echoed the millet system of the Ottoman Empire. The Cypriot would be a citizen of a single sovereign Cypriot state but internally would be a citizen in a bi-communal state and a voter of an ethnically divided electoral college. This aimed to provide representation to positions already divided according to specific quotas, according to the Constitution and an example of consociationalism.

In consociationalism "pure" parliamentary systems do not work because they cannot secure the fixed percentages of brokered power between the two communities. Either because the population could change or districts gerrymandered, because of internal division among the communities etc. Usually this is resolved by bi-cameral institutions, checks and balances between the branches, judicial review etc

The cypriot solution was for the President to be Greek, the vice-president Turkish, the assembly comprised by 35 Greek and 15 Turkish parliamentarians etc. Still the government was unitary and it was expected that consensus politics would be required, given the veto power of the turkish community. Of course this never came through given the animosity both between the communities but most importantly their mother states.

So the answer is that it became a Presidential republic because it was better suited to accommodate a unitary government of a bi-community consociationalism state while not devolving into two federate parts. The division across ethnic communities of the electoral college aimed at providing representation according to pre-arranged percentages and a Presidential system allowing for "strong men" was better suited for a unripe democracy divided according to ethnic communities loyalties rather than a common cypriot identity. These "strong men" like Archibishop Makarios had already a long history and were accepted leaders of their community, while fostering relations with Greece/Turkey/UK.

Sources: a seminal work is H.Richter's "History of Cyprous" but also interesting is an insiders view of the Zurich-London agreements and the disagreements in greek diplomatic circles between the greek minister Averof and future Nobel laureate Seferis in "Αβέρωφ-Σεφέρης. Η ρήξης". Still, I am not familiar with any turkish literature on the matter and would gladly see any insight as how Turks envisaged the matter.