in petasō can't mean "on", only "inside". The "on" in the dictionaries is an artifact of the English usage where the distinction between various prepositions is often vague or associated with particular semantic fields, or even individual words.
That's what a Latin speaker would say, any way, because in Latin there's absolutely no overlap between the word for "inside" in and "on top of" super, although there is an overlap between the latter and "above" suprā. You're always in viā, forō, aedibus, lūdīs, scholā, īnsulā, terrā, while in English you're "on the island, mainland, Earth" but "in the building, some country, in school", and either "on the street, forum" or "in the street, forum"... but "at the lesson". Just google for "English on vs in" to find scores upon scores of confused learners.
'Dī bonī!' - exclaims a confused Latin speaker - 'how can in terrā be both "in some country" and "on mainland/Earth" (not to mention "in/on the soil"), and in scholā both "in school", "in the school" and "at the lesson"? And how in Earth is in spectāculō, lūdīs both "at the concert, games" and "in the concert, games"?? And what's that business with "street" and "forum"??' What can one say but shrug? :-]
You could of course try explaining to them that the choice of preposition depends on how the space in question is conceived of by the speakers, and whether it's actually a space or a time referred to metonymically with a word for a space, but that would be like explaining to an English speaker that the choice of perfective vs. imperfective verbs in Russian depends on whether one conceives of the action as a point or a stretch on the timescale. 'But what if I just want to say in scholā???'