It's kind of hard to distinguish the usage, for the most part "onto" is used when there's direction involved and "on" usually implies stationary location, but that's not entirely accurate. Sometimes they are interchangeable. However, for some reason it feels more natural to say "put something on the table" rather than "onto" even though there is some sense of the object moving from one location to the other.
This happens with a lot of instances of "put": "I put a new leash on my dog." "He put a bandage on the wound." "I put my left foot on the ground."
So you are probably right that "put X on Y" is idiomatic English.