Post-synaptic vs pre-synaptic inhibition
Yes, inhibitory post-synaptic potentials (IPSPs) are always in the context of post-synaptic inhibition, because they are post-synaptic potentials. They occur because of inhibitory neurotransmitters (for example, GABA) are released and bind to post-synaptic receptors, particularly ligand-gated chloride channels. We often just call this "synaptic inhibition."
Pre-synaptic inhibition, on the other hand, would reduce the frequency/amplitude of excitatory post-synaptic potentials, by reducing neurotransmitter release at excitatory synapses (for example, glutamatergic synapses). You can also have pre-synaptic inhibition at an inhibitory synapse, where the pre-synaptic inhibition is actually disinhibitory from the perspective of the post-synaptic cell (inhibiting inhibition).
In summary: post-synaptic inhibition is reducing the rate or probability of action potentials; pre-synaptic inhibition is affecting the quantity or probability of vesicle release.
Interneurons vs inhibitory cells
Inhibitory interneurons are a major class of interneuron in the neocortex, but not all interneurons are inhibitory (layer 4 spiny stellate cells, for example, are excitatory interneurons). In addition, not all inhibitory cells are interneurons; some brain regions have inhibitory projection neurons, for example the cerebellum (Purkinje cells) or striatum (medium spiny neurons).
Inhibitory cells are those that release inhibitory neurotransmitters, so they are best thought of as involving post-synaptic inhibition. Chandalier cells in cortex are a bit of a special case, because they synapse onto axons and interfere with action potentials that way, but I would not lump them in with presynaptic inhibitory mechanisms.
Sources of pre-synaptic inhibition
In general (because there may be exceptions, and almost certainly are in invertebrates who have very "weird" nervous systems from my mammal-biased worldview), presynaptic inhibition arises from three places: autoreceptors (self receptors), retrograde signalling from post-synaptic cells, and neuromodulation. One could fill textbooks with information on all the mechanisms, but I'll give one example of each of the three types:
Glutamatergic synapses, for example, can have pre-synaptic inhibitory metabotropic glutamate receptors. These are G-protein coupled receptors, not ion channels, and they typically respond when a synapse is very active. They are effectively a brake on over-activity: if a cell is firing too much, the effect of its firing will be decreased by pre-synaptic inhibition mediated by these self-receptors. This principle is common for other neurotransmitters, too. (Wu & Saggau, 1997)
Presynaptic inhibition by post-synaptic cells
Although we think of neuronal signalling as one way, that's not entirely true. Post-synaptic cells have mechanisms to communicate with the pre-synaptic cell, and this can include inhibiting that cell.
Endocannabinoids are one mechanism: they are released by the post-synaptic cell and can reduce pre-synaptic release probability (Maejima et al 2001; Melis et al 2004). At an excitatory synapse, that makes them inhibitory; at an inhibitory synapse, they would be disinhibitory.
Pre-synaptic inhibition by neuromodulators
Pre-synaptic cells can also be affected by local concentrations of neuromodulators, like dopamine (Bamford et al 2004). Dopamine isn't necessarily released directly onto a cell but more into the surrounding area. Dopamine D2A receptors, for example, are found presynaptically at cortico-striatal synapses. Release of dopamine in the striatum reduces glutamate release at those synapses coming from cortex.
Bamford, N. S., Robinson, S., Palmiter, R. D., Joyce, J. A., Moore, C., & Meshul, C. K. (2004). Dopamine modulates release from corticostriatal terminals. Journal of Neuroscience, 24(43), 9541-9552.
Maejima, T., Hashimoto, K., Yoshida, T., Aiba, A., & Kano, M. (2001). Presynaptic inhibition caused by retrograde signal from metabotropic glutamate to cannabinoid receptors. Neuron, 31(3), 463-475.
Melis, M., Pistis, M., Perra, S., Muntoni, A. L., Pillolla, G., & Gessa, G. L. (2004). Endocannabinoids mediate presynaptic inhibition of glutamatergic transmission in rat ventral tegmental area dopamine neurons through activation of CB1 receptors. Journal of Neuroscience, 24(1), 53-62.
Nadim, F., & Bucher, D. (2014). Neuromodulation of neurons and synapses. Current opinion in neurobiology, 29, 48-56.
Wu, L. G., & Saggau, P. (1997). Presynaptic inhibition of elicited neurotransmitter release. Trends in neurosciences, 20(5), 204-212.