Not sure if it's the preferred term or not but the cover for the business end of a pen is often referred to as a lid. Either term would be acceptable in the UK

Not sure if it's the preferred term or not but the cover for the business end of a pen is often referred to as a lid. Either term would be acceptable in the UK though. We generally don't make the distinction. We also colloquially sometimes refer to a bike helmet as a "lid" too. Although that is slang.