We present color-magnitude diagram analysis of deep Hubble Space Telescope imaging of a mass-limited sample of 18 intermediate-age (1 - 2 Gyr old) star clusters in the Magellanic Clouds, including 8 clusters for which new data was obtained. We find that ${it all}$ star clusters in our sample feature extended main sequence turnoff (eMSTO) regions that are wider than can be accounted for by a simple stellar population (including unresolved binary stars). FWHM widths of the MSTOs indicate age spreads of 200-550 Myr. We evaluate dynamical evolution of clusters with and without initial mass segregation. Our main results are: (1) the fraction of red clump (RC) stars in secondary RCs in eMSTO clusters scales with the fraction of MSTO stars having pseudo-ages $leq 1.35$ Gyr; (2) the width of the pseudo-age distributions of eMSTO clusters is correlated with their central escape velocity $v_{rm esc}$, both currently and at an age of 10 Myr. We find that these two results are unlikely to be reproduced by the effects of interactive binary stars or a range of stellar rotation velocities. We therefore argue that the eMSTO phenomenon is mainly caused by extended star formation within the clusters; (3) we find that $v_{rm esc} geq 15$ km/s out to ages of at least 100 Myr for ${it all}$ clusters featuring eMSTOs, while $v_{rm esc} leq 12$ km/s at all ages for two lower-mass clusters in the same age range that do ${it not}$ show eMSTOs. We argue that eMSTOs only occur for clusters whose early escape velocities are higher than the wind velocities of stars that provide material from which second-generation stars can form. The threshold of 12-15 km/s is consistent with wind velocities of intermediate-mass AGB stars in the literature.
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