Answer: C is correct, I was late to dinner; there was a traffic jam.
Explanation: The semi colon signifies a pause, which when you speak this sentence outloud, there is a slight pause.
The first one is a run on
The second in the exam does not have a comma, its "I was late to dinner with a traffic jam." and is clearly illogical because a traffic jam is not a person you can go "with"
The third is correct
The fourth is speaking of the jam as if its a concert or event. Which is incorrect.