We demonstrate fully transparent, highly rectifying contacts (TRC) on amorphous GaInZnO and compare them to TRC fabricated on single crystalline bulk ZnO and heteroepitaxial ZnO thin films. The contacts' transparency in the visible spectral range exceeds 70%. From numerical simulations, we conclude that thermionic emission is the dominating transport mechanism, however, for several samples with low net doping density diffusion theory must be applied. The detailed investigation of the rectification properties of the TRC using temperature-dependent current-voltage and capacitance-voltage measurements reveals that barrier inhomogeneities govern the IV-characteristics of all diodes irrespective of the sample crystallinity. Assuming a Gaussian barrier height distribution, the extracted mean barrier heights typically range between 1.1 and 1.3 V. The width of the barrier distribution correlates with the mean barrier height and ranges from 110 to 130 mV. By compiling literature data, we found that this correlation holds also true for Schottky diodes on elemental and III-V semiconductors.