We reconstruct the two-dimensional (2D) matter distributions in 20 high-mass galaxy clusters selected from the CLASH survey by using the new approach of performing a joint weak lensing analysis of 2D shear and azimuthally averaged magnification measurements. This combination allows for a complete analysis of the field, effectively breaking the mass-sheet degeneracy. In a Bayesian framework, we simultaneously constrain the mass profile and morphology of each individual cluster assuming an elliptical Navarro-Frenk-White halo characterized by the mass, concentration, projected axis ratio, and position angle of the projected major axis.. We find that spherical mass estimates of the clusters from azimuthally averaged weak-lensing measurements in previous work are in excellent agreement with our results from a full 2D analysis. Combining all 20 clusters in our sample, we detect the elliptical shape of weak-lensing halos at the $5sigma$ significance level within a scale of 2Mpc$/h$. The median projected axis ratio is $0.67pm 0.07$ at a virial mass of $M_mathrm{vir}=(15.2pm 2.8) times 10^{14} M_odot$, which is in agreement with theoretical predictions of the standard collisionless cold dark matter model. We also study misalignment statistics of the brightest cluster galaxy, X-ray, thermal Sunyaev-Zeldovich effect, and strong-lensing morphologies with respect to the weak-lensing signal. Among the three baryonic tracers studied here, we find that the X-ray morphology is best aligned with the weak-lensing mass distribution, with a median misalignment angle of $21pm 7$ degrees. We also conduct a stacked quadrupole shear analysis assuming that the X-ray major axis is aligned with that of the projected mass distribution. This yields a consistent axis ratio of $0.67pm 0.10$, suggesting again a tight alignment between the intracluster gas and dark matter.
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