Your British colleague is “on holiday”. Your American client is “on vacation”. Same beach.

German has one word for this: “Urlaub”. English has two, and which one people use depends on the English they learnt.

British English: holiday = Urlaub. American English: vacation = Urlaub.

In Global English, the English of international business, both are understood everywhere. You don’t need to choose sides.

One thing worth knowing: in British English, ‘holiday’ can also mean a public holiday (Feiertag). “I’m on holiday” means you have time off work. “It’s a holiday” means the office is closed.

Quick win: Use whichever word feels natural. Both are correct.