Well, there are some pretty big differences, and you are looking at a description of Republic used exclusively in the United States.
A Republic is a nation where the government of that nation is considered a "public affair" (literally translates from the Latin phrase "Res Public" which means "Public Thing") as opposed to a Monarchy, which is an inherited form of government (The King's first child shall be the next leader).
A Democracy is a government that has some measure of elections.
Not all Democracies are Republics. The United Kingdom and Japan are both Democracies but have a regent, a Queen in the United Kingdom, and an Emperor in Japan. These are Constitutional Monarchies which means they are functionally similar to Constitutional Republics, in that the laws are voted on in some fashion, but their head of state is a Monarch who is governed in what he or she can do by the Parlimentary Body, headed by a Prime Minister, who is the head of Government. In Republics, a President will be elected to fill a similar role as the Monarch, with Semi-Presidential (i.e. France, Germany) having explicit powers shared with the PM and Presidential (i.e. United States, Mexico) having all Head of Government and Head of State functions performed by the President.
Similarly, not all Republics are Democracies. As @origimbo, North Korea claims it is both these things, but to the casual observer, it is none of those things. The real answer is a bunch of Legal Fiction in that, Kim Il-sung is still the elected leader of the nation, and his son (Kim Jong-Il) and then Grandson (Kim Jong-Un) are holding down the fort until Kim Il-sung returns (yes... from the grave). Why yes, I am aware of the joke that the more adjectives in a title promoting freedom, the more like a prison the place is.
Other famous Republics that are not Democracies (or rather, Liberal Democracies, like the U.S., U.K., Canada, Japan, Switzerland, and a handful of many other nations) include the Soviet Union (The Union of Soviet Socialists Republics, officially) and pre-Iraq War Iraq. They both touted their elections, but neither was what most Western Democracies would call elections. The USSR was a one party state, so all people running were from the Communist Party and Iraq had a periodic question for election that was basically "Do you want Saddam Hussain to continue being President? Yes or No" and had surprisingly little support for No (aided by the arrest of all no voters). Normally these kinds of governments are Illiberal Democracies. Offering people the ability to vote "Like they do in the United States" is a great way for a modern dictator to hold power because the population has a concept of voting, but not what an actual ballot looks like.
Finally, when the United States says "It's a Republic, not a Democracy" it should be pointed out that Republic can be substituted for a "Representative Democracy" and Democracy can be substituted for "Direct Democracy". This goes back to the fact that the United States is generally one of the first Representative Democracies in the modern world and much of it's constitution was copied extensively by other nations making such a transition. At the time, Representative Democracy wasn't a real phrase, and the United States took government inspiration from the Roman Republic and Greek Democracy in their creation of their government model. The Founding Fathers were rather cagey about Mob-Rule from Direct Democracy and thought that a Republic was a better system to allow the people to vote, but to not vote in favor of insane polices... Look at the name of Boaty McBoatface for a more tame self rebuke on pure Democracy.
Even up to the point of Lincoln's Presidency, Democracy was strictly Direct Democracy where as Republic was a Representative Model. Generally they also held that a Republic was required to protect those with unpopular opinions (i.e. the minority political party) from getting outlawed and worked for equal rights to vote for representation (Republicans were the party of both the African American vote and pro-Woman Suffrage in the United States. This is before the party flips of the 60s, 70s, and 80s)