You are counting infinite regions which are not bounded:
Those have no definable "area" as the question pertains. Thus I claim you are wrong and even the designers of the question are wrong if they expected your answer.
The question says "their intersecting lines form 1,597 areas".
That doesn't suggest finite areas to me, or to most mathematicians.
If finite areas were meant, it would have been made explicit in the question.
Your solution should start out by stating that you interpret area to mean finite areas.
Anyway, getting back to my question:
What is the maximum number of finite areas created with L lines?