Excerpt from the german constitution:
Democracy means rule of the people by the people. The people exercise government power in elections and have the last word in monitoring the activities of key government institutions. These are the five "constitutional bodies", i.e. the Bundestag and Bundesrat with responsibility for legislation (constituting the legislative branch of government), the Federal Constitutional Court with responsibility for supreme court decisions (constituting the judicial branch of government), and finally the Federal President and the Federal Cabinet with responsibility for executive tasks (constituting the executive branch of government). The Cabinet is responsible for the conduct of government business.
You see even constitutions in real live mix the definitions here.
A republic is rule of the people by the people. This means representants are elected and decide.
A federal republic is a republic with locally partly independent areas (provinces), forming an additional body for the state. The areas make local rules, which can be overridden by national rules.
A democracy is the rule of the people. In fact, there is no democracy today. Look at the DyP mod, where the real governmental types are implemented. Democracy was there before republic. The federal republic is the "most modern" kind of government. The problem what was received with a true democracy in ancient ages was the time taken for decission was too long. Thats why the Republic was invented.
Now to our game:
What we have now is clearly a republic.
We elect people, they decide for us. Thats fact. The only advantage the demogame has is that we have democratic (?) poll-rules. But we have almost no measure for leaders not discussing and publically polling their decissions.
As we saw in the last terms, almost no leader decission was polled about.
We do not have a federal republic, because we have almost no local politics in our game. We have provinces make build queues, but thats it. We are on the way to federal republic though, as maybe someday in future we will allow local elections and local laws. The senate was a first step to it.
A true democracy would try to poll almost all decissions on the pupulace. Only inevitable things would be decided by the president.
I really hope your republic game will get us a full federal republic with full local politics.
But that arises another question:
Why dont we implement it in here? It will be almost the same, just that we get local politics.