Meaning of "projection" and "to project" in neuroscience
I think the definition you found from OED is fine. Dictionary definitions are often not really ideal for terms used in biology, you learn more from examples and usage, for example the quote you gave about upper motor neurons.


  can I say that a neuron projects to another neuron, to mean that the axon of the first connects to some dendrite of the second?


Typically yes, though as the OED definition implies, "from any given"/"to the receiving part", projection is typically used for longer-range connections, where the sending and receiving cells are in different nuclei or layers. You would be less likely to describe cells that synapse on each other within one nucleus as projecting, though I also wouldn't say that description is entirely wrong either, and the distance required to be "long-range" can sometimes be short.

You'll note if you go through the Purves' glossary that you linked, all the uses of "projection" refer to long-range connections.