Haha, in situations like this one and the Democratic Republic of the Congo, sometimes, playfully and incorrectly of course, I use the term "Ρεπουμπλικανική Δημοκρατία"! I just love it how Hellenic "Democracy" and Latin "Republic" work so well together in so many different languages, but somehow, in Hellenic itself, this doesn't seem possible!
The president of the Republic is indeed correct but I put “The president of the democracy” and it was incorrectly voted correct. I couldn’t think of an alternative, the correct sentence would be “The president of the democratic state” but I though you wouldn’t accept that!
This is something I have struggled a lot to understand in English myself. For example, the UK is considered to be a democracy while it's not a republic. It seems that people are happy (or maybe it's some sort of linguistic compromise) to take the (ultimate) power away from the people and use the term democracy as a placeholder for all other freedoms/powers the citizens of such a country enjoy. It really makes the mind boggle. If anything, the word democracy should be used as the exact synonym of republic and some other word should be used for states where people have some powers. Or, you know, have republic for the more general term since it says nothing about who actually holds the power.
@mahdaeng If you check the etymology of democracy (and the only way of interpreting this word in Greek is attached to its etymology) you'll see why Greek speakers find it hard to separate democracy from republic.
Well, for me a republic is led by a president - such as Macron in France or Trump in the USA. Most democratic countries are also republics, but not all. The UK is not a republic because it is a monarchy and we have a queen, but it is a democracy. A republic must have a president. A democracy for me is a country in which the government is elected by a democratic voting system - although this is claimed by some countries which are actually dictatorships as mahdaeng says below. Very interesting, though, that the words seem indistinguishable for Greek speakers.
I like your suggestion that a republic is a representative democracy. That is fairly accurate. A democracy is, in general terms, the rule of the people. Sadly, even mob rule can be labeled a democracy. A republic is a representative system in which agents are chosen to represent the will of a group of people. As you stated, it is a representative implementation of democratic ideals. Thus, in some cases, a government can be considered both republican and democratic simultaneously.