First I thought it was just the British. They have car terminology that is different from the Americans. Like petrol vs gasoline, boot vs trunk, gearbox vs transmission etc etc. But during my quick search I found this page:
https://www.greenandwhiteauto.com/services/outside-the-car/spring-headlight-replacement.html
They appear to use the term headlight for the whole light, and headlamp specifically for the actual bulb inside the headlight… I find that a little odd. Or maybe they just want multiple search hits, just like suggested above.