Sorry—double post!
The night-night megathread has illustrations on where to press. Supposedly packing a solid, blunt point into the neck wrapping right over the carotid helps. I tried cut raw potatoes. The right size, shape, and firmness, but too slippery. I couldn't get them to stay.
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My understanding is that the goal is to restrict blood flow in the carotid artery, which will bring about unconsciousness and death very swiftly. The problem is that the carotid is pretty deeply embedded in the neck tissue. Most people seem to end up crushing their windpipe and/or jugular veins instead. This can kill you, but it's a lot more uncomfortable. It's also slower, which increases the risk of either being found and "saved" or reflexively thrashing yourself free in the window between unconsciousness and death. Brain damage is a likely result.
I tried the "night-night" method, which is quite similar to partial-suspension hanging, and never managed to do more than compress my jugulars. I could feel my head filling up with blood, which is a very nasty sensation. There's got to be a way to make partial work, but most people seem to give up on it. (I did.)