Ah

This is true for takeoffs but not for landings. You want to land with a headwind because this means that for the same airspeed you have a lower groundspeed, i.e. when you actually touch down you're going slower on the runway than if you touched down at the same airspeed but with a tailwind.

In other words, it's because of the greater lift.

No. Your calculated landing speed doesn't change depending on the winds, so you'll always be touching down at roughly the same airspeed for the same aircraft type, weight, and flap setting. Touching down with a headwind just means you don't need to use the brakes as hard.

It's beneficial on takeoff because headwind already factors into your airspeed before you even start rolling, which gives you more lift for the same groundspeed yes, enabling you to rotate sooner than you otherwise would.

You're literally repeating what the other guy said with different words.