Touched down is what the plane did. The plane is the subject of the sentence and is where the focus of the sentence is. Put [the plane] down is what the pilot did. The pilot put the plane down on the wrong runway.
However, there has long been one situation where the plane put down is idiomatic. That's when put down is a euphemism for an emergency landing. In this case, the pilot had, shall we say, less than full control of the plane. Longman has this to say under definition number 10 of "put down" —
10 AIRCRAFT put (something) down if an aircraft puts down or if a pilot puts it down, it lands, especially because of an emergency
The engine failed and the plane put down in the sea.
So if you want to be persnickety, you can make the argument that put down is an acceptable answer, but it's not the best answer.