I wonder if the lithium wake up time has to do with the change in internal resistance as a function of temperature. The graph below shows Li internal resistance going up much more sharply than lead acid at cold extremes(well, at least Li-Ion, not some of the other Li chemistries). But I don't think that necessarily means the battery capacity is reduced, it's just that it can't put out as much current for a give voltage charge. Then with an Li battery, hitting the starter and then waiting a few seconds warms up the Li battery, lowering its internal resistance, and allowing to supply more starter current?