As to the positional relations of two lines L1and L2 in the plane, we classify them into the following three :
L1L2 =
Parallel (Not intersecting)
L1L2 Intersecting (Not parallel)
L1L2 = {M} for a point M
L1 = L2 Coincident
Remarks
"Coincident"
Here we understand "coincident" as follows :
The line which L1 indicates coincides with the line which L2 indicates.
We need not bring such a troublesome problem as :
"What are L1 and L2 when they are in fact the one thing?"
"Intersect"
The mathematical wording of "intersect" is somehow different from the daily one.
In mathematics, "intersect" is used in the sense of "holding a point in common".
Especially, "coincidence" implies "intersecting".