Quotes from p6 of your first reference below. It is very clear that the ice core temperatures are regional.
"As these water molecules are evaporated, primarily from the oceans, the lighter molecules, those having fewer neutrons, are preferentially evaporated over the heavier ones, due to a slight difference in vapor pressure caused by the extra neutrons. This causes the vapor to be depleted in heavy molecules but enriched in lighter ones. As the air mass cools and condensation occurs, the heavier molecules preferentially condense due to the same principle The condensation is then assumed to fall out of the cloud as precipitation. Thus, the oxygen isotopic ratio of rain and snow is strongly related to condensatIf the temperature of the air mass should continue to fall, the condensation will contain decreasing concentrations of the heavy molecules, resulting in a depletion of 18O relative to precipitation that condensed in a warmer environmention temperature. . ....In the context of ice cores, this technique allows scientists to estimate the actual air temperature of condensation when the snow fell".