I hear Americans use the term liberal all the time but the way they use it makes me think we're talkibg about different things

A liberal is someone who:

1. Upholds the modern nation state and is thus against monarchy (against whom the first liberals rebelled against)
2. Upholds capitalism and market economies, and with it property rights
3. Upholds electoral parliamentary systems of governance
4. Usually believes in some version of the social contract or similar theory from which the legitimacy of the nation state and capitalism is derived.

Anyone from the left complaining about liberals is using this definition of liberals (typically). The basic reasoning for using this definition if liberal is that it has always been the definition of liberal and has only changed recently in some parts of the world. It is also not necessary to change the definition because the "progressive liberals" also mostly fit the old definition either way. Pretty much every serious socialist political theory will start with a criticism of the philosophy of liberalism.

It does. Literally the president of the United States as a prime example.

The conflict in terms comes from what in Europe mostly describes the social axes. Social liberalism is very different from what in America usually refers to economically neo-liberals who are basically late stage capitalists