From what I understand:
L1p: made to work ok with 1.5v alkaline batteries which means it has to keep operating down to 0.8 volts or so since alkaline batteries' voltage sags as they run down. Not terribly efficient and not tightly regulated. Efficiency and regulation at such low input voltage is quite difficult to accomplish in such a small space. Not sure what happens if you try to run it at 3 volts--maybe it blows up.
L2p: made to run on two AA alkalines, tightly regulated and somewhat more efficient than L1p, but doesn't work at such low voltage. Can operate down to 1.2 volts or so which means it works ok on one AA L91 lithium (1.7 volts) or NiMH (1.2v with flat discharge curve) with better performance than L1p on those types of cells, though still somewhat out of regulation. But with an alkaline cell (most of the energy available from an alkaline is at below 1.2 volts) it performs quite a bit worse than the L1p.
L1T/L2T: these are the same circuit and the new design works better at low voltages than the L2P circuit, though maybe still not as well as the L1p on 1xAA alkaline (I'm not sure of this). Anyway since "low" mode is available, worse efficiency on 1xAA alkaline is at least partly alleviated.
L1D/L2D CE: I don't think anyone has seen or tested these yet. L0P CE's are starting to appear and there's been some informal reviews but no real measurements so far.
