One big difference is that in the U.S. cardholders are largely protected from credit card fraud (not debit card fraud), so the card vendors have to take the risk and so have robust anti-fraud measures (both before and after payment). Largely it is the merchants who have to prove that there was no fraud. Whereas in Europe the burden of evidence (not proof) is with the cardholder.
You get nervous about giving your card to a waiter because you’re in a foreign place with a nonsense payment system worst than most developing countries and it’s not something you’re ever asked to do anywhere else.
In the US, you simply have no choice if you want to eat in a restaurant, so people are used to it. I'd expect total skimming rates to be higher in the US, since magnetic stripe transactions have been phased out in effectively all other countries. People don't care because they don't directly pay for the resulting fraud out of pocket. As a society, of course everybody still pays for it.
> Largely it is the merchants who have to prove that there was no fraud
No, in-store, it's the issuing bank that's liable, even in the US (unless the card is PIN-preferring, which is usually only true for foreign cards).
There's also a large difference between counties. In the Nordics its ubiquitous, I haven't carried or needed cash for almost 20 years. Meanwhile Germany has barely started to use cards.