Today these distinctions are rapidly disappearing and on-onto and in-into are used indiscriminately. We read and hear sentences like these all the time. “He got in his car.” “They went in the hotel.” “He climbed on the horse.” “The passengers got on the plane.”
While I am in favor of shorter words and the simplification of English spelling, I don’t like these changes. I consider them to be lazy grammar. They’re inaccurate and sometimes confusing. Using these words incorrectly reflects badly on the speaker or the writer and does little to improve communication.
When I read “he jumped on his horse,” I have a mental picture of a man jumping up and down on the saddle on his horse’s back. (And what does the horse think about this jumping on his back?) When I hear “he got in his car,” I ask myself what did he get in his car? A sandwich? Typhoid fever? Mugged?
But the change has already started, and you may disagree with me. Perhaps you see no great problem and even think that the change is quite acceptable as long as there is no ambiguity. And perhaps you may be right.