I would never consciously, deliberately hit a child as a punishment. Firstly because it's crap parenting, and secondly because it's illegal in the country where I live. I have never heard a parent threatening to hit their child here and only ever once seen it happen (in nearly 14 years of having children). In that one case a father slapped his 2yo son who was throwing sand, and every other parent in the sandpit looked aghast. Nobody called the police or anything drastic, but there was clear public disapproval.
I've also never heard of a parent here having a child removed by SS for being smacked - only for more serious violence as in the UK. The social disapproval seems to be enough in most cases to deter people. It possibly goes on in private - hard to judge how much - but the fact that this constant violent undercurrent is absent makes for a much more pleasant atmosphere generally. People in the UK seem to have become immune to the violence and the threat of violence simply because 'smacking' is tolerated. Hitting your DC is lazy parenting.
The favoured method of discipline for a small child here is to kneel down to their level and say very firmly, but quietly, 'you do not do this, you do not throw sand/bite others/whatever, it is wrong' IMMEDIATELY after the misbehaviour. Depending on the age and behaviour you could then either leave the group straight away or leave if it happens again. If the child is old enough I would also get them to apologise to the other child.
We never used a 'naughty step' and I believe that too is a British concept. It's obviously unworkable for younger children, but we have used a form of time out when the DC were older. DD1 went through a phase when she was 8 or 9 of misbehaving/provoking at the dinner table, at which point we asked her to leave so we could eat in peace, and she had her food alone later. She was old enough to understand why she was being punished and that the punishment was a logical consequence of her behaviour. We have also used 'no TV' etc as sanctions for older DC. Now they're 13 and 10 often a raised eyebrow or the Death Stare or The Voice is enough to keep them in line.
Having said all that, yes of course I've totally lost it at times when the DC were younger. I've shouted at them, and on occasion I have grabbed a wrist to separate them fighting or pushed one of them away from the other when verbal warnings didn't work. I admit at that point I failed - physical intervention would ideally not be necessary. What I don't do is try to justify it as some sort of good parenting - it's actually just evidence that I am as fallible as most of us. I have also apologised to the DC afterwards for my actions and explained that violence is never an answer to the problem.