"First Hellenic Republic" is a historiographical term used by academics the actual name of the government was the Provisional Administration of Greece and was established at the beginning of the greek revolution and later renamed as the Hellenic state. There was no separation between head of state and head of government.
Provisional Administration of Greece (Presidents of the Executive, 1822–1827)
The convention offered the throne to Prince Otto of Bavaria. They also established the line of succession which would pass the crown to Otto's descendants, or his younger brothers should he have no issue. It was also decided that in no case there would be a personal union of the crowns of Greece and Bavaria. Otto went on to rule Greece until he was exiled in the 23 October 1862 Revolution.
In October 1862, King Otto was deposed in a popular revolt, but while the Greek people rejected Otto, they did not seem averse to the concept of monarchy per se. Many Greeks, seeking closer ties to the pre-eminent world power, Great Britain, rallied around the idea that Prince Alfred, the second son of Queen Victoria and Prince Albert, could become the next King. British Foreign SecretaryLord Palmerston believed that the Greeks were "panting for increase in territory", hoping that the election of Alfred as King would also result in the incorporation of the Ionian Islands, which were then a British protectorate, into an enlarged Greek state.
The London Conference of 1832, however, had prohibited any of the Great powers' ruling families from accepting the crown of Greece, and in any event, Queen Victoria was adamantly opposed to the idea. Nevertheless, the Greeks insisted on holding a referendum on the issue of the head of state in November 1862. It was the first referendum ever held in Greece.
The Second Hellenic Republic was a parliamentary republic which was proclaimed on 25 March 1924 and a referendum was held to abolish the Monarchy. The Second Republic was abolished after the 1935 monarchy referendum. The President of the Republic had a symbolic role as head of the state.
On 1 June 1973 the junta abolished the monarchy and replaced it with a presidential republic. The abolition of the monarchy was approved by a rigged referendum held on 29 July 1973.
On 24 July 1974, the junta was overthrown and democracy restored. A second referendum, held on 8 December 1974, confirmed the abolition of the monarchy and the establishment of the current parliamentary republic, with the President of the Republic in a symbolic role as head of the state.
^There weren't any formal political parties then. According to Greek historian Grigorios Dafnis : "...it is not possible to talk about parties during the revolutionary period" («Υπό αυτούς τους όρους δεν είναι δυνατό να ομιλούμε περί κομμάτων κατά την περίοδο της επανάστασης και μέχρι της 3ης Σεπτεμβρίου 1843»). «Τα Ελληνικά Πολιτικά Κόμματα 1821–1961» Εκδ. Κάκτος, σελ. 190 (2020) Grigorios Dafnis: "Greek political parties 1821–1961", Athens:Kaktos, p.190