It's possible that you didn't get faint because you didn't press long enough. I'm told that intense pressure isn't necessary, so it shouldn't be that. It's also possible you were mostly pressing on your vagus nerve instead of your carotid. The branches of the vagus nerve run through the same general area on the sides of your neck as the jugular and carotid. Massaging the vagus nerve stimulates your parasympathetic nervous system, which pretty much triggers the reaction you describe: tingly and euphoric. It won't kill you, but it might make you want to die less. That's definitely worth something.
The carotid seems to be really hard for most people to hit on themselves … supposedly the place you want to hit is this Y-shaped fork where the internal carotid artery meets the external carotid artery and becomes the common carotid artery. The PPeH describes putting your fingers right at the angle of your jaw, and then moving a little bit down into the soft tissue of your neck. Your jugular is right in there too, and depending on how you're put together, it might be hard to isolate just one structure.
If you've mostly got the jugular, it will make your head feel like it's going to explode. The vagus nerve kind of gets its little feelers wrapped around everything in there, which is fine because pressing it or massaging it can feel great. Pressing on that V shape in the carotid is supposed to make your vision start graying out pretty fast. I've never managed to hit the stupid thing, but I assume it's similar to losing consciousness for any other reason. Vision gets dark around the edges, you might feel lightheaded or sick, and then
Goodnight Moon.
There's a lot more discussion about this in the Night Night Method thread, if you haven't read it yet.