That one would be appropriate here, and would not imply finiteness.
I think a majority of people would answer 4 if you ask for
the number of areas formed by 2 intersecting lines on a plane surface.

You're right my question is no longer relevant...

Btw, the answers to the two questions differ by the number of unbounded areas,
which is easily seen to be 2*L; one for each line-end.