OK so your situation is complicated, but I like a theoretical question so I will go with a theoretical answer. Take this in the spirit it is meant 
(No I'm not NT either
)
The current research evidence that we have suggests that punishment has an extremely minor role in discipline/behaviour change. It's interesting because you'd assume that it is THE most effective thing. But apparently it's not - punishment on its own does not cause a long term change in behaviour.
What most people mistake for the effects of punishment are most often two things - one is threat. If somebody is afraid of you, and it's not very difficult as an adult to wield power over a child and make them afraid of you, then this triggers self-protective mechanisms and they will probably do what you tell them to do - so you get compliance in the moment. Or at least, you get an appearance of compliance in the moment. But this is compliance - so your child is doing what you say, and if you do that enough then they might begin to automatically do that and form a habit, but they aren't necessarily internalising this as a value or thinking/learning about why to do this/not to do this, or what to do instead. Also, when they think you won't find out, they are may well go and do the behaviour anyway. Children (especially older children) can also learn to lie and sneak very well in order to avoid being caught doing the behaviour.
There is a reason that basically the only people seriously advocating for this kind of parenting today (where the aim is to make the child genuinely fear/dread the punishment, and punishment is the main form of control) are strongly religious, because if you believe in a punitive, omniscient God, then it's a way for that threat to continue, both out of direct sight of the parents, but also into adult life. Otherwise it stops being effective once the parent is no longer a credible threat. Laws basically work like this as well - there is a difference between crimes which are backed up with social norms where you're likely to get caught (stealing, violence) vs crimes which people feel are arbitrary - downloading films, speeding. Most people don't feel bad doing those things if they think there is no harm caused and they won't get caught.
The other thing that usually is responsible for behaviour change when people cite punishment as the catalyst is that most parents aren't only relying on threat of punishment - they do all kinds of positive things as well, for example you might punish children for fighting, but you also praise them for sharing, plan activities which are more structured and less scope for fighting, model and prompt problem solving and apologising, talk about examples of good conflict management and so on - the vast majority of the behaviour change is supported by all the non-punishment stuff, and punishment is not actually necessary, although it can serve to emphasise the message and speed up behaviour change, so it's not entirely true that it does nothing at all. It does something. However, also surprising about this - it doesn't matter how severe the punishment is. You would assume that effectiveness scales with severity, but it doesn't. In fact, there are two things with this - first of all, if you get too severe/physical/intimidating, then you start going into threat territory, and this has some quite serious consequences if the threat is persistent, in that children can develop disorganised attachment and/or trauma if their parent is both attachment figure and the source of regular threat, especially if it's unpredictable and attached more to the parent's moods than any particular behaviour expectation. You know all the research they have been citing for decades now about not hitting children? It turns out that the harm from smacking is now thought to be more likely to be related to the unpredictability/chaos/physical threat rather than the action itself (which doesn't mean smacking is OK FWIW).
Then the other issue with punishments being too severe is about length/duration - for example a screen ban - if you make it too long, then it totally loses efficacy because the child essentially ends up in a scenario where they never get screen time in the first place so they have nothing to lose.
So ideally, in order to maximise effectiveness and minimise any negative effects, you should only be using very minor, short-lasting consequences, and you have to balance them with plenty of positive input. Also, the parent should be calm and in control, which usually means drawing up a behaviour plan in advance and using punishment (alongside all the positive stuff which will actually be the driving force of behaviour change) in a targeted way to change specific behaviours, rather than just sort of having it in your back pocket as a way to get children to comply in the moment. (Recognising BTW, that almost everyone does this and it's not necessarily a problem to do this as long as it's not causing undue distress and you understand it's not a long term way to change behaviour).
Interestingly the idea that is currently very popular about natural/logical/related consequences is not really evidence based and is usually counterproductive because most people misunderstand the point of what natural consequences are. That is a tangent so I won't go into it in detail but suffice to say, if you've decided in advance what merits a punishment and you want the most minor/token thing, then something generic is usually easier and more predictable, rather than having to figure out a related consequence in the moment. In the moment is a bad time to decide on a punishment because you'll generally be at least irritated, at worst furious, adrenaline-fuelled and exhausted - and the evidence shows that people come up with more severe punishments when they are even slightly irritated. So it can help you to stay calm and controlled and fair, if you know in advance what the punishment for certain behaviour is.
Oh - there likely is a third role of punishment as well which is communicating social disapproval - that taps into our inherent desire to be accepted/follow social norms. That tends to work the best when the person on the receiving end is getting the same disapproval from multiple sources so for example if a child at school is told off by a teacher for some behaviour but all of their peers think the behaviour is very cool and approve of it, the punishment is unlikely to deter. But if e.g. a younger child gets the message that stealing is wrong, and this is reinforced by several sources e.g. teacher, parents, peers (e.g. in play) then they are likely to absorb this message.
In terms of books - there are lots of books but not all books contain good advice
if you're genuinely interested in the theory of how punishment works there are three resources I'd suggest:
Alan Kazdin - American professor of psychiatry and child psychology - he explains things really well and he is very very big on the positive opposite side. He has a free course on Coursera which is run through Yale university called "The ABCs of Everyday Parenting" which is sort of behaviourism explained in a practical way for parents. This is actually really good and I would recommend it to everyone even (especially!) if you're sceptical about behaviourism.
123 Magic - a book which is older, but basically explains the same principles. I don't like this one personally, because I think it's much too focused on compliance and it has a really large focus on time out, but it is evidence based and is probably the best-known modern(ish) parenting book which does include punishment. Also, I found it unnecessarily graphic/upsetting that the book included descriptions of the ways that parents had previously abused their children before finding this "magic" method. I think that was included in order that you realise whatever you're feeling guilty about, it's not as bad as that, but I just found it disturbing to read and couldn't get the images out of my head. I say modern-ish because it was published in 1995 and it shows. The info by Kazdin is more up to date, and you may notice one aspect which is that 123 Magic starts with how to punish unwanted behaviour, moves into how to incentivise preferred behaviour and has connection as a sort of afterthought - whereas Kazdin's method starts with how to identify and encourage a preferred behaviour using a combination of incentives and connection, and the section on punishment is almost the afterthought, and he keeps hugely emphasising the effectiveness of the positive aspects. The focus on punishment is why I included 123 Magic here, but it's interesting to contrast how two applications of the same theory approach this in different orders and what that means in terms of what parents get from it.
Who's In Charge - book and course by Eddie Gallagher who is a psychologist and therapist from Australia. The courses are run by several UK local authorities, focusing on parents whose children are violent towards them. Honestly the book is in desperate need of an editor. It's far too long and most of it is his very meandering and not always well-informed opinion, which seems to boil down to "Kids today have no respect and parents make too many excuses!" There are a few interesting observations, but not enough to slog through 500+ pages. The valuable part starts at the chapter called "Acceptable and Unacceptable Behaviour" and from then on it goes through using the author's extensive experience with families to explain what is helpful and what traps parents can fall into and how to avoid them. This one has much more granular details like exactly how to use things like pocket money fines and how to respond in the moment when the behaviour is occurring. I think it can be quite helpful but I definitely wouldn't see this as a general parenting guidebook.