What is the difference between pre-synaptic versus post-synaptic? As a non-specialist, I am unclear as to the exact meaning of the terms pre-synaptic and post-synaptic.

Specifically, do they refer to 


the same neuron, either transmitting or receiving, before and after the chemical synapse has occurred, OR
two different neurons, i.e. "pre-synaptic" = the transmitting neuron, and "post-synaptic" = the receiving neuron?


I ask this because the terms are often used in ways that do not make it clear which meaning is the one intended.
Typically 'presynaptic' and 'postsynaptic' are used to indicate two neurons that are connected, as you indicate correctly in your second example. Information flow in the nervous system basically goes one way. If one neuron fires (presynaptic cell) it can chemically activate another cell on which it synapses (the postsynaptic cell), as shown in the following figure 1.   

  

As an illustrative example consider the auditory system (figure 2). 



The cells that send their axons from the inner ear to the cochlear nucleus (the first central auditory structure in the auditory pathway) are called spiral ganglion cells. The axons from the auditory nerve cells form the auditory nerve. The auditory nerve cells release glutamate from their axon terminal into the synapse, that in turn activates the cochlear nucleus cells. In this scheme, the auditory nerve cells are presynaptic, and the cochlear nucleus cells are postsynaptic. 

Translating this example into Figure 1, the axon on top would be the auditory nerve cell and the postsynaptic neuron the cochlear nucleus that receives input from the auditory nerve.

However, to continue with the peripheral auditory system as example, the auditory nerve cells receive their input peripherally from hair cells in the inner ear, while the cochlear nucleus cells send their output centrally to the olivary complex. Hence, auditory nerve cells can also be classified as postsynaptic, while cochlear nucleus cells are also presynaptic. Again, as illustrated by this auditory-system-based example, it all depends on the context and the specific system under investigation. 

The terminology can, however, point to a single cell as well. In the hypothetical scenario a neuron would send a collateral axon to its own cell body, it would be the presynaptical as well as postsynaptical cell. In the figure, the axon on top of the image (still correctly labeled as presynaptic) would then come from the neuron shown on the bottom of the image (still labelled correctly as being postsynaptic).