In this scenario, there is a particular regime wherein the punishment for any and all violent crime is the same- a single gash to the neck.
Upon pronouncement of guilt by the crown or a recognized officer of the crown, a warden will place a small but extremely sharp dagger against the convict's neck, apply just enough pressure to slightly dimple the skin, then rapidly withdraw it along its own axis.
Here's the twist- the victim, or a close family member if the victim died during the crime, may instruct the warden on the exact placement of the dagger. If the victims feel generous or forgiving, they will pick a position unlikely to cause lasting harm; if they want vengeance, the dagger will be positioned elsewhere. In any event, the convict is free to leave immediately after the punishment is administered, including to seek medical attention.
The warden may not advise the victim, but if you know the right people to talk to, anyone can learn the concepts beforehand. Experience from prior punishments has made it part of the collective knowledge that if you position the dagger just so, there will be a lot of pain but the convict will most likely recover in a few days. Move it around just a bit, you will sever a certain nerve and cause partial paralysis. Move it up or down a notch and the convict will bleed out from the jugular long before he can get to help.
Is this idea plausible? In other words:
Does human anatomy include distinct positions along the neck that can result in a. mere temporary pain, b. partial paralysis, and c. rapid death, given a single shallow gash at just that right position?
Can these points be easily taught to lay people?
Other assumptions/ background, if needed:
An accused's trial includes both determination of guilt, and whether the crime was violent enough to warrant this sentence.
Lesser crimes involve lesser sentences, most of which involve corporal punishment in other forms
Feel free to use the most liberal definition of "neck" as you wish
Technology level is early industrial (i.e. mid to late 1700s Earth)
EDIT: Here is a very rough doodle of what I have in mind, both a top view and side view: