In most of the world, Red is the color used by left-wing parties and movements, while Blue is often the color of conservatives. In America red is typically identified with conservatism and the GOP, while blue is identified with progressives.

It didn't always used to be that way. During the 1970s and 1980s, the major news networks started using color coded election maps, and there was little consistency between the networks. In 1976, ABC used yellow for Ford and blue for Carter; CBS and NBC during the 1980 election used blue for states that Reagan carried and red for states that Carter carried.

Because red was associated with communism and the USSR, both parties wanted to claim blue as their color during the Cold War. Once the Cold War was over that connotation diminished. It wasn't until the 2000 election that the networks decided to all be consistent with each other. Here's a great article from Smithsonian Magazine outlining the history of how those colors were used to identify American political parties.

In the rest of the world, Liberal refers to typically conservative pro-business parties, while in America this is the opposite.

One thing to understand about terms like "liberal" and "conservative" is that their meanings and ideological implications are dependent on the historical and national context in which they are used. It gets confusing as those terms both imply directionality (i.e., liberalism is associated with change; conservatism is associated with preserving the status quo or returning to tradition) as well as ideology (i.e., liberalism is associated with greater freedoms; conservatism is associated with fewer freedoms). So, for example, who would have been the "conservatives" in the USSR during the Cold War? Were they hardline communists who resisted the progressive policies of glasnost and perestroika? Or were they traditionalists who sought to return the USSR back to its monarchial state? As you can see there is some ambiguity in how we use these words.

When "liberal" is used in Western European nations, it generally refers to "classical liberalism," which as you mention is associated with free market economics. In the United States, liberalism came to be associated more with social issues during the 20th century. On top of that, the liberals who advocated for equal rights and fought against institutionalized racism also advocated for economic policies that favored government intervention such as the New Deal and the War on Poverty. So that's why liberalism has a different connotation in the United States—it's a reflection of different progressive policy positions that were largely shared by one voting block.

In the rest of the world, a Libertarian is essentially a Socialist or Anarchist, while In America, it refers to an ideology of pro-business and privatization.

I'm not aware of any place where libertarian is used interchangeably with socialist (feel free to correct me), but libertarianism is definitely closely related to anarchism, and is more synonymous with minarchism. The difference being, anarchism advocates for no government at all, while minarchism advocates for small government. If you advocate for small or no government, that's essentially an argument for privatization and free markets.

The whole Red=Conservative/Blue=Liberal dynamic in the States is rather fascinating too. As noted, the U.S. TV networks began using varied color schemes in election maps back in the '70's. But it wasn't until the 2000 election (Gore/Bush) that they really took hold.

The reason being NBC political reporter Tim Russert. According to The Washington Post, Russert appeared on NBC's The Today Show about a week before the election to go over the projections for the race. Just happened to be on that day there was a color-coded map that was Red=Republican, Blue=Democrat. Russert asked how George W. Bush would "get those remaining 61 electoral 'red states', if you will?"

The election would turn out to be highly contested, eventually requiring a Supreme Court decision (Bush V. Gore). The case dragged on for months and the terms "Red State" and "Blue State" caught hold in the media, and later for the American public, illustrating the stark cultural and political differences between some of the states.

Another point is that in other countries the left-wing parties that use red are quite similar, starting from labour movements in the late nineteenth and early twentieth centuries (Labour, UK 1900; Labor, Australia, 1912;Labour, NZ, 1916; the French Socialists sort of count too, though they use a kind of pink and the current form of the organisation dates to the 1960s). These parties were associated with Socialism, and picked red themselves. This movement never really took off to the same extent in the US (a Socialist got 6% of the vote in the 1912 presidential elections, but that was about as far as it got) and the left is represented by the Democrat party, which didn't have those same associations