This is largely a linguistic quirk whose validity isn't a very interesting debate topic. Language does strange things sometimes.

> Curious... I always read 'America' when people from USA talks about them.

Not just americans talking about the USA. Most of the world. From Britain to Japan, america is used to denote the US and americans to denote citizens from the US.

> What about the people from the rest of the continent?

North Americans ( for US + Canada ). South Americans for people in south america. There are even Central Americans for everyone in between.

In a geographical sense, mexico is also included when people talk about North Americans ( like NAFTA for example ). But from a cultural or general use, north americans primarily mean americans and canadians.

'America' is only used in the English name of one country: USA. When talking about countries, using English, America is shorthand for the USA. Americas on the other hand, would mean all of the N/S American continents.

Canadian here.

It's simply because "The United States of America" is a mouthful so everyone either uses the last word "America" or shorten it to "USA".

"US" can sometimes be confusing because then you are not sure if people are saying "us" the object pronoun or "us" the country", so "USA" it is.

There are several different schemes for dividing the continents. There can be as few as four or as many as seven, depending on who you ask.

English-speaking cultures generally agree that there are two continents that lie entirely in the western hemisphere, North America and South America (collectively "the Americas"). In that scheme, "America" without modifier clearly refers to the USA.

Romance-language cultures generally agree that there is only one continent, America. In that scheme, calling the country "America" looks like chauvinism. But it's really just a linguistic difference.