Umbra, penumbra and antumbra
The calculations seem reasonable. The conclusion is wrong.

The Earths antumbra starts 1.3 Gm from Earth. At less than 1.3 Gm there is umbra, penumbra and non-shadow. The moon is 0.4 Gm from Earth. 

$$0.4 &lt; 1.3$$

Hence it is impossible for the moon to be in the Earth's antumbra.

However, the Moon's antumbra forms at almost exactly one lunar distance. We notice that the moon and the sun are almost exactly the same size in the sky. When the moon is further from the Earth, an annular eclipse can form, as a small part of the Earth will then be in the moon's antumbra.

I've made a scale model of the sun-earth-moon shadows, at https://ggbm.at/CbbvmBFq