People have self described as classical liberals for quite some time. It's more to do with the curious development of liberal coming to describe just the American left when the traditional sense of the word would describe most of the American political sphere.
Indeed, we need a translator for US to European political terminology. Note, terminology, not parties as the Democrats have a range of people from centre left (Warren etc) to centre right (Yes, Obama's overall policies were centre right on any normalised political spectrum)
US on the left, Europe on the right:
Liberal - Progressive, Left Wing. See the UK Labour party at present, Podemos, Syriza
Centrist - Right Wing, Obsession with hating the left usually a factor here. The Tory Party, Republican party in France, and CDU in Germany are equivalent. All have their climate change sceptical wing too.
Because the US is a 2 party system, the actual Centrist people will be Democrats. There is no equivalent of the Liberal Democrats, for example. Socially progressive, and usually a mixture of economic views but mostly biased towards free markets. Because of that, actual Centrist views are lumped in with "Liberals" and thus, pushing the political spectrum rightwards as the right wing try to claim the centre as theirs.
However, most centrism parties in Europe are firmly anti fascism because of the impact it had here. And while the Lib Dems would gladly tear Corbyn a new one, they would never ever pull the both sides bullshit that US "Centrists" would say.
Nor would they treat dog whistle politics as anything other than what it is.