Churchill's comment "this is a situation up with which I will not put" comes to mind (on his being confronted by a pedant wanting to change Churchill's sentence to stop it ending with a preposition). To see the myth put to bed firmly, read Fowler. Both the current Burchfield and the previous Gowers edition show how the myth came about, as well as dismissing it as inconsistent with good English. It's like the idea that you shouldn't start a sentence with an "and" or split an infinitive - things you sometimes learn as gospel in school (though not from teachers who truly know their English), but which are actually much more fluid in the best English. And I would second the idea that "to put a coat on" is perfectly acceptable English. "To put on a coat" can imply that you mean a coat rather than a jacket, so changing the position can change the meaning.
It's also the feel and the sound of it -- a sentence in any language rings true , or not, to native speakers, if they've learned their mother tongue well to begin with. Both are acceptable English. One is better, more fluent English. If you want to evoke pointless pedantry your assertion that the order of "on" in this sentence changes the meaning (??) is a good example.