Yes, there's a difference in position. In Swedish, you can ställa, sätta or lägga, which all correspond to put in English.
Things that you ställer (past ställde) end up in a standing position.
Things that you lägger (past lade or la) will lie down
Things that you sätter (past satte) are in an intermediate position that can be thought of as "sitting", or the position afterwards is less clear.
So if you ställde flaskan, the bottle will be standing up afterwards, but if you lade flaskan, the bottle is now lying down.
No, it's not as common in English any more - I just thought it would clear up some confusion to make a direct comparison. "Put" has only been used in the current sense in English for a few hundred years - before that its meaning was closer to "push". Instead, words like "lay" were used. Both the English "lay" and the Swedish "lägga" come from the same Old Norse root "legja". "Ställde" has the same root as the English "stall" - but obviously the meaning has changed in English.
American English speaker here, and I actually answered "he stood the bottle on the table", and it was marked wrong. To me, that's an entirely normal way to say it, just as it would be to say "he laid the book on the table", or "he set the computer on the chair" (I would use "set" here, not "sat"). In fact, I probably don't use "put" terribly often, and certainly not more often than I would use the more descriptive words.
In Swedish you do seem to use them more than in modern standard school english , but my Grandad rarely "puts" things anywhere. He sets something down, lays stuff on the table, or "stands it over there"...or even slings it in a corner. I've never seen him "put" a thing anywhere in my life! Then again he never "turns" a corner either, he swings, hangs or rounds it. XD
Now I think of it, It is amazing the depth of language we are losing in the commonly spoken modern standardised english :( as i use these terms far less often, and often use a much less rich vocabulary to describe common actions than he does.
Gee, I'm a native speaker and I didn't know that "rule" . For me, onto and on are pretty much completely interchangeable. Onto should be accepted, but let's not complicate things by saying the translation "should be onto the table" implying that the current translation is wrong. Edit - when an action is implied. I wouldn't ever say the picture is onto the wall, for instance.