I think you are right. British/Irish/Australian/NZ/SA people are far more likely to be familiar with American words and phrases than vice versa.
It's important to understand that US English is not a 'foreign' language for us. Most of us are exposed to American English from a very early age, starting with Disney cartoons and moving on to Hollywood movies, US TV sitcoms, American music and so on. As English speakers we all grow up with a passive knowledge of American English to such an extent that hearing American words and phrases seems completely normal to us. Young people in Britain and other English-speaking countries particular are essentially bilingual when it comes to listening skills. For example, a small child in Britain understands when they hear that the American family in their favourite US show takes the 'elevator' up to their 'apartment', but they will talk about taking the 'lift' up to their 'flat' in a British context.
By contrast, Americans have far less exposure to non-US English. Believe it or not, some British TV programmes for children are actually 'translated' into American English and redubbed for US audiences!