I notice that if I hold something in front of but somewhere between my eyes, even something in the middle will be further displaced from the center of my field of vision (assuming that's what it's caused) from closing one eye than from closing the other. Why is that?

Some people have a dominant eye, but for some they are equally dominant. If you have a dominant eye, you normally see everything from the angle of that eye. If you close that eye you then look at the object from the angle of the other eye. This will change both where the object is in your field of vision and what parts of the object you can see.

The effect you are describing seems to be parallax, which is caused by the fact that we have binocular vision. This effect is used in part to give humans a sense of depth.