The chain-saw grooves remained a great hit with all the birds. I was sad to leave it behind when we moved from California to Iowa, but that birdbath was simply too heavy to take with us. Since then we've used birdbaths of metal, ceramic, or cement and found all these materials satisfactory. No deeper than three inches at the center. It should be even shallower at the edge, so that a bird can ease its way in.
Many commercial birdbaths are too deep.
If you already own a deep birdbath, you can put rocks in it to raise the bottom, though this will make it a little harder to keep clean. The plink of falling water is pure invitation to birds. It dramatically increases the number of species that visit a birdbath. For example, hummingbirds would never wade into the bath like other birds, because they bathe only in flight. But I have watched hummers zipping back and forth through the drips, timing their flights so that they catch a water drop on their backs on each pass.