I think the name L is chosen because it corresponds to the 'L operator' of 'old algebraic geometry' over the complex numbers (and, and that in turn is named L, I would agree, probably to honour Lefschetz). I don't know what notation Lefschetz used himself, but for example Griffiths-Harris book "Principles of Algebraic Geometry" introduces this L operator on page 111 in the deRham setting. Also as the key player for the Lefschetz decomposition/Hard Lefschetz.
The whole book makes no reference to motives or Weil conjecture stuff, just down to earth complex manifold stuff. So, just as a reading suggestion, if you want to read about it on an elementary level.
Maybe this is an easier starting point than things over finite fields etc., even though admittedly maybe not very sexy.
The L operator here wedges with the Kähler form, and that would be just an explicit (1,1)-form representing the point in P1.
(by representing I mean that it represents the cycle class Chow -> deRham of the point via an explicitly given differential form - and that should come down to representing the 'motive' of it)