grammar was #onething that happened

A teacher-trainee asked me this question :

“I imagined him winning the game” vs. “I imagined his winning the game”
Which is more common or makes sense?
In the first sentence, is ‘him” an object for the verb or a subject for the gerund? In the second sentence, is ‘his’ a subject for the gerund?
As you know, Korean students tend to analyze a sentence, and some asked me the questions above. My answer was just ‘”Both may be grammatically correct.'” Was I right or wrong?

This was my answer:
I think you were correct in saying that both are grammatically correct. Of course we know that “him” is an object pronoun and that “his” is a possessive adjective This means that they become the object complement.
However, it also seems that both of these can also act as the subject of the gerund. However, it did take a lot of research for me to be able to tell you this. According to the link I shared, this can be found in Cambridge Grammar of the English Language.
I hope this gives you some peace of mind! :) Let me know what you think.
That was the best answer I could give, and it took me a while to formulate it. I share this with you as my response to Anne Hendler’s #OneThing blog challenge because these are the types of questions I get from Korean high school teachers of English, and I struggle to answer them. I struggle because they ask me to go to linguistic depths that I don’t usually dive into. I know my strengths as an English teacher lie more on the sociolinguistics side. However, these questions are very important reminders of the reality of my teacher-trainees. They keep me aware of the types of challenges they face.
Like the teacher, there is still a little doubt in my mind as to whether or not my answer is satisfactory. But that’s another reason why I’m posting this: it’s time to air out the doubt, and face my grammar demons. If you have any thoughts to add to this grammar question and my attempt at an answer, I’d love to hear them.