Republican - The REAL DEAL

The first political party in the United States was known as the "Republican Party". But, we can't even use that name anymore without creating confusion. We generally call the first "Republican Party" (the faction led by Jefferson and Madison) the Democratic Republican Party, because the modern Republican Party stole their name. The modern Democratic Party is actually the continuation of the original "Republican Party".

In an even earlier case, the same political faction had taken the name of Whig Party, since most Americans still fondly remembered that most American Patriots of the Revolution were "Whigs" while they were still part of the British Empire. It was like a bunch of Tories calling themselves Whigs to confuse voters.

In more modern times, we have the Tea Party who have taken a name originally given to bunch of Whigs who gathered to protest Corporate Monopolism and taxation without representation.

There's also the ongoing attempt to redefine the word Liberal in the US.

So, what it all really means
What it means is that too many people in the US really are confused about some of the language used and understood in the rest of the world. Many of us, who aren't Republicans, understand the meaning of republican (note the lower case letter "R") in the common sense of the word, and we are republicans (again the lower case letter "R") in the common sense. But, we have to be careful using the word because Republicans (note the upper case letter "R") are constantly trying to baffle us with bullshit or are so baffled by the bullshit they've been listening to on Fox that they don't or can't understand common sense.