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.