El Niño‐Southern Oscillation (ENSO) strongly influences the interannual variations of Madden‐Julian Oscillation (MJO)—the dominant mode of tropical intraseasonal variability. Previous studies have revealed more about the impacts of the diversified El Niño on MJO, but less about those of the La Niña diversity. Using observations and global climate model experiments, this study demonstrates that MJO undergoes distinct variations and Pacific teleconnections when La Niña varies between eastern‐Pacific (EP) and central‐Pacific (CP) types. Compared to the CP La Niña, MJO under the EP La Niña is more energetic over the southern Maritime Continent–South Pacific Convergence Zone and can propagate faster and further with a broader zonal circulation scale over the Indo‐Pacific warm pool. The stronger and faster MJO under the EP La Niña is associated with more vigorous ascending motions, higher moisture content and stronger moistening in the lower‐level atmosphere ahead of the MJO deep convection. Although the meridional moisture advection exists as the fundamental mechanism of MJO eastward propagation, the distinct MJO under the two types of La Niña primarily results from the different lower‐tropospheric moistening due to the zonal moisture advection. Both the background mean and anomalous zonal winds and moisture gradients are stronger under the EP La Niña, thereby causing a remarkable invigoration effect of zonally advective moistening. Diverse extratropical circulation responses of distinct MJOs are also discussed. These results should improve our understanding of the whole ENSO diversity paradigm that influences MJO and teleconnections.