The form and severity of the punishment itself is irrelevant to the determination of when corporal punishment changes into abuse.
For the record, this entire answer pertains to your own children. There are some situations where some of this can extent to others that you are responsible for ("Thanks for watching the kids, sis. Tim has been punching Sue recently; if he does it again feel free to give him one swift pop on the butt. See you in a couple hours."), but the answer is meant only for your own children.
As I stated in a comment, my father sometimes spanked me harshly enough that I could feel it a day or two later, and his father was even more harsh with him. But this, or worse, does not necessarily constitute abuse in and of itself. Please bear with me here and hear me out...
TL;DR: Don't use physical punishment when you are angry, and do it only out of love when actually necessary.
I know the rest below is a lot to take in, but I filled it with honest truth and love, so I hope you can get through it all and with an open heart.
An abuse outline
What really matters is (feel free to come up with your own outline, but I think mine is pretty good):
Was the punishment just (as in "justice")?
a. If not, it might be abuse and you need to reconsider it.
What were the parent's motives?
a. Bullying your children is often disguised as punishment; that is abuse and is evil. Spanking your child out of anger often falls into this category; if you are angry at the child you should think twice (or thrice), and possibly refrain even if you're telling yourself they deserve it.
What damage was done to the child?
a. Lasting physical injury is almost certainly abuse, and actions which have the potential for permanent injury are definitely abuse.
b. Acting in a manner which damages the child psychologically is abuse.
The list could go on; feel free to make your own, but try to be honest about it instead of just saying "It physically hurt the child so it must be bad."
In the absence of such signs of abuse, there probably is none, especially in the presence of obvious wrong-doing from the child.
My household example
I generally don't discuss this with others, but this seems important and like it might be well received by you.
I do use physical punishment with my children. I prefer to use it as sparingly as possible, and I try to set a situation up to avoid it if I can, but if I feel it's necessary it will happen.
Generally, physical punishment is used when one or more of my children are being blatantly disobedient (they are already being bad, they are given a direct order in an effort to improve the situation, they refuse), or if they continually do the same thing even after a large number of gentler disciplines in a short time, or if they do something severely bad (like "Wow, my mouth just dropped open because I can't believe you just did that" bad).
When I do use physical punishment, especially if it was severe, I often stay with the child after to make sure they understand why it happened, and sometimes to provide hugs or cuddles while telling them that I love them. Sometimes I genuinely cry with them, because I can't stand hurting them.
Understanding and love
Above all else, I try to make sure that children understand as fully as possible why they are being punished or disciplined, what was bad about their behavior, and how they could love others more. That last one is key in our house: love.
We make sure that all the children understand love and hate. If my oldest son does something purposefully to hurt his sister or she says something mean to him, all we have to do usually is say "Are you hating each other?" ("Yes dad. [with frowns] I'm sorry .") "Do you want to hate each other or love each other?" ("Love.") and then they start to do nice things for each other.
Maturing from understanding
Because I make sure my children understand love, hate, good, and bad, they usually can tell the difference. And because lies are not tolerated at all, and we go more lenient on offenders who tell the truth, they usually come out with the whole truth and explain the bad things they did in even more detail than we knew.
When we have the opportunity to discuss like this with the older children who knew precisely what happened and "spill the beans," this allows us to take it to the next step, which for me is to ask them what types of disciplines or punishments are appropriate for their actions. Sometimes they cite spanking as an appropriate punishment for a certain action.
If my children are truthful, and if they come up with a descent list of appropriate disciplines or punishments that fit their offence, I often let them choose which one will happen. This right here was my main reason in wanting to do this answer, to explain this part: sometimes my children choose the physical punishment over a non-physical alternative. No matter what they choose, if it gets to this point, I try to provide the smallest version of the punishment possible, and sometimes even provide ways for them to get out of it entirely; at this point, they don't need instruction and only need punishment if the offense was severe.
Note on severity
As said earlier, the severity itself does not necessarily imply abuse. There have been a few (very few) times I have spanked my children very harshly, nearly as badly as my father used to with me, but these times were always because of very dire need.
The best example is with one of my sons who used to do things that were actually dangerous - not to himself, but to his brothers. When I am worried that one of the children could cause terrible injury to one of the other children (or to anyone), then I would - and should - do whatever is necessary to squash that behavior. No other disciplines or punishments were working, and the health of others was riding on it, so out came some very severe physical punishments... and they worked.
Note on maturity of the parent
If we want our children to respect and honor our decisions, one of the best things we can do is to set a good example. I strive to set an example to my children of how to love each other. I also admit to them when I'm wrong, even as it relates to discipline or punishments.
Sometimes I make a mistake in the disciplines or punishments that I dish out. When that happens, I feel terrible. I want my children to be able to tell me freely when that has happened, and sometimes they do. Sometimes they do not, since disrespectful arguing (which it often turns into) is itself punishable, but if they are respectable about it they can talk to me about what has happened. And sometimes I ask them later about the incident.
If a punishment I provide is over-reacting, or worse was in error, I apologize to my children. This one really gets them to respect me more. Nothing turns their anger into understanding (or acceptance) better than when I talk to them about whether my action was just or unjust. Usually they admit that what they got was just and deserved. Sometimes they insist that I am not understanding why they did something and that they had no ill intention and deserved nothing negative; when this happens I apologize to them and ask them to please forgive me, which they always do (and I often try later to secretly do something good to make up for it best I can).
Another note an parent maturity: Whether it's with punishments or other things, I try usually to guarantee their situation is never worse than my own, so I know what they are going through. For physical punishments, this often means that I take no precautions to protect myself, and indeed sometimes I specifically make it worse for myself. Often, if I take physical action, I am in at least as much pain after, often more pain. If one of my children tells me that I hurt them too badly, I have a few times done to myself whatever I did to them and made sure that I got it worse than they did... this has caused me to apologize to them before and ask for their forgiveness if I did not realize, for example, that even a light smack on a certain spot hurts disproportionately much.
I know that I do not abuse my children when I use physical punishment since I try to avoid it at all costs until I feel it is actually necessary, I do it out of love, and I admit when I was wrong.
Essentially, if you are acting out of hate then it is abuse, but if you are acting out of true love then it is not, and you can perform a hateful act even against people that you otherwise love.
The physical punishment itself is not abuse. What is abuse is when parents hurt their children out of anger instead of love, when they hurt them with lasting injury (physical, emotional, mental, or spiritual), or when they are bullies.
Show them mountains of unconditional love, and then once your children are old enough talk, very gently, to them about it openly.