Canter is used in the US as well, but yes, here in Italy it's called galoppo (gallop). The two phrases are "in piede" and "l'inforcatura." I'm fluent in Italian so the problem isn't understanding the meaning of it, I was just curious as to the purpose of being "in piede" when cantering as I don't remember doing it when I was younger. But now I'm pretty sure based off of things my instructor has said that it's just to help with getting the body trained to keeping heels down and working on balance for when we do jumps. But let me know what your neighbours say! My instructor doesn't speak English so I would be interested to hear it from their perspective of knowing both languages and being riders.It would be helpful if you posted the two Italian phrases. We speak some Italian and also live next door to Italians who ride ride horses.
In case it has not been said, the term Canter (relating to Canterbury pilgrims) is not used outside the UK. The footfall of the horse is different in Canter as opposed to Gallop, but some languages use the same word and some dont have a specific word at all.