It’s great that you have had to correct the tense. It really shows how the language works.

Question 1. I see now how much more elegant your version is, “I was sitting at my desk playing”. Yet if you left my awquard version, would you say ”I was sitting at my desk. I was playing a game” or “I was sitting at my desk. I played a game”? I suppose it would be the former “I was sitting at my desk. I was playing a game”.

Question 2. If your final version is “I was sitting at my desk. I was playing a game” how would you describe “I sat at my desk. I played a game.” Can the misuse of the tense be called a rude mistake or is it the way people often say that? Can this with the Past Simple be said by your neighbor, a bank clerk, your president or a teenager chatting in a bus? I would like to feel the degree of mistake.

Question 3. Are you sure that “I fought with Peter today” is absolutely correct? Could the boy have said “I have fought with Peter today”?

Now that you have corrected my opus, Kevin, I deeply feel how important the correction of what one says is. I must have read and heared this pattern “I was sitting playing” a hundred of times, yet I have hardly payed much attention to it. As you saw I can easily go with my awquard “I was sitting and playing”. After correction it somehow gets curved ... no, not curved... carved? or stuck? or set? (Help me, Kevin) in the person’s mind, I guess.