Even our own world doesn't have just one type of republic. You have:
- Presidential Republics, with an executive presidency separate from the legislature, such as the United States, most of Central and South America, parts of the Middle East and Central Africa.
- Semi Presidential System, with both an executive presidency and a separate head of government that leads the rest of the executive, who is appointed by the president and accountable to the legislature, such as Russia.
- Parliamentary system, with a ceremonial and non-executive president, where a separate head of government leads the executive and is dependent on the confidence of the legislature. Republic of Ireland and India, Germany and Finland.
- Republics with an executive presidency elected by the legislature, South Africa.
- Constitutional Monarchy, with a ceremonial and non-executive monarch, where a separate head of government leads the executive. The UK, Spain.
Beyond that, you're going into full monarchies, one party countries and military governments.
Point is, a republic isn't very specific, its an umbrella term covering quite a few different forms of government that share some key aspects.