Yes, you are correct that there is only one L1 for any pair of bodies. Gbaikie is not saying anything different. He is saying that there are two pairs of bodies which include the Earth:

1. Earth / Moon
2. Earth / Sun

Both of these pairs of bodies have L1 points. Those L1 points are different. The Earth is included in both of those pairs of bodies. Therefore, when one is only making reference to the Earth, and not the second body, you can say that it has two entirely different points that can legitimately be called "L1". Only when you are talking about a pair of bodies can you say that there is a single L1 point.

Oh, and it was Lagrange, not Laplace.

Cheers,

Nathan.