I see the MN parenting experts have stepped in all with their conflicting opinions about what you should get in your time machine and change. 
The fact he's on referral for ADHD assessment is hugely significant, and nothing you've said about your parenting sounds like a problem at all. Come on over to the SEN boards if you want to. I also have a younger one who probably doesn't have ADHD, but has totally picked up on some of the more successful attention-seeking behaviours of his older brother. It is very very hard. It won't hurt the younger if you approach both of them the same way. It's essentially normal parenting anyway, just a bit more careful and specific.
Big punishments won't work for ADHD - punishment has very little effect at all really. Nothing reactive/in the moment really does. Proactive parenting is the key, meaning you have to be one step ahead of them at all times. As you can probably appreciate, being one step ahead of child who is constantly at 100mph is exhausting! But the more you can do this way, the smoother life goes.
You can do reward/points charts as some people have suggested. Actually, removing certain privileges by default and letting them earn them back is a very good approach but it has to be manageable for them to achieve something. If you set the bar too high, then they will easily get discouraged and stop trying.
If you are going to do punishment, and it can be useful to have something "in your back pocket" for when you're tempted to slip into scolding or engaging in an argument - it needs to be something extremely small and token, like losing a point, or a 5p fine, or losing 5 mins screen time, or a short time out (we do 20-30 seconds per year of age for the ADHD child). Don't do long punishments which linger into the next day, don't do big dramatic things (like losing ipad for rest of day) that can only be used once a day, never use anything permanent as punishment (e.g. cancel birthday party). ADHD children tend to find being punished/told off very upsetting because it feels completely arbitrary to them. By keeping the punishment mild, it helps them get into a mindset where they can accept OK, I did that thing wrong and now I have a consequence for it, but I can handle that - and they ALWAYS get another chance to try again the next day/time/whatever. This is incredibly important because they can have such an outsized emotional reaction and really lean into "I'm bad/I never get things right/this is the most unfair thing ever", but you don't want them wallowing in that kind of thinking, you want them to reflect on their behaviour and try not to repeat it.
This is confusing because the usual assumption about punishment is that the more upsetting the child finds it, the stronger the lesson/the more likely they are to avoid that behaviour again, whereas this isn't strictly true, and it's very unlikely to be true with ADHD. The punishment won't change the behaviour anyway - it's everything else you're doing around this which will help change the behaviour.
Don't worry about making consequences related/logical. It doesn't matter. They won't register any difference at all. The only exception to this is that sometimes it makes sense to point out a natural consequence, e.g. this toy got broken because you didn't put it away - and sometimes restorative action is helpful in showing them the impact of their behaviour (e.g. helping put something right, helping pay for a ruined/lost item, doing something nice for someone to apologise.) If you need to change a boundary because of behaviour (e.g. reduce access to fizzy drinks) don't present this as a punishment/consequence.
In the moment, the most important thing is for adults to stay calm and for responses to be predictable and make sense - so it can help to work out the rules in advance and write them down, possibly even consider holding a family meeting so the DC can have input as well. You can also sometimes de-escalate in the moment by using things like humour or distraction.
Outside of the moment, looking at scenarios where you can immediately predict there will be problems is incredibly helpful and helps you come up with your proactive game plan. Sometimes there are issues due to a lack of skill or difficulty with something e.g. if a child is easily distracted, it might not be a behavioural problem, it might be that they genuinely find it difficult to "screen out" unrelated stimulus, so there might be accommodations you can make to the environment or tools they can use to help with this e.g. noise blocking headphones, or being sat in a part of the classroom where they are less easily distracted. Or if a child is always creating instead of cleaning their room as they have been asked, it's possible they are having difficulty because they don't know where to start and feel overwhelmed, so it can help to build that skill with them by scaffolding it (e.g. breaking it up into smaller tasks).
Siblings fighting may be communication or conflict management, so you could explicitly work on these skills. If I don't know what skill is behind a behaviour challenge, I've found it's actually usually a reasonable starting point to ask ChatGPT - then whatever answer it gives me, I will go looking for more reliable sources of information but it's normally right. Sometimes it suggests something I'd never even have thought of, like cognitive flexibility - but when I go and read up about what that actually is, it makes total sense and then I can usually find ideas of different games or activities to try to practice/develop that skill. (This is how I find myself playing elaborate pretend games with the DC where we swap roles and become a customer vs shopkeeper who is variously anxious, angry, easily upset, bouncy etc)
For things like taking them to a restaurant or into a shop, it can help to remind them of the behavioural expectations explicitly before you enter - decide what they should be in advance and think about issues you might come up against and prepare for them, e.g. bring an activity to do while you wait for the food which is not likely to inspire sibling wars or too much physical enthusiasm. Or look at the menu at home before leaving so they have chance to decide in advance what to eat and ordering can be as quick as possible.