op I really feel for you. There are so many times when I have been the embarrassed parent with my ds, for different reasons maybe, but I recognise that feeling. On the other hand my dds are like the ones in your post!
My ds is amazing, very sensitive and emotional, which can mean very loving, passionate about things, like cruelty and caring. But it can also mean that EVERYTHING is such a big deal at times, and he will go off in one at the drop of a hat. He can throw the most unbelievable tantrums, and he doesn't care where he is or who is watching. He is now 9, and it took me a long time to switch my head round.
At first I was in the place you are in, worried about what others think, that sounds shallow, but it comes from the desire for other people to like and love my ds like I do. I had to really consciously make a decision that my reaction to him had to be always and totally in relation to his behaviour, and nothing else, no matter who was watching or how horrid he was being. I had to take my reaction out of the equation. Forget the worry that your parenting skills are being judged. THEY are not parenting your ds. I see parents with 3 lovely laid back dds and think they don't know they are born!
I aree with what others have said too, boys can be very different. They need action, don't do sitting still in a cafe until they are about 18 school age, but especially they react physically. He he wants a toy, he will grab or snatch. At 3 he may even bite or hit. A girl in most contexts would ask, whine, scream or shout. Of course you have to respond to the physical, and you have to give him the words, so ' when you want a toy you need to say, can I have a turn next please, not bite' My ds was an early talker and very chatty, but he still does reacts physically before verbally.
It does get better. The sleep gets better too, because he is getting to the age when you can say don't come to me, if you wake up you are a big boy and can do a wee and go back to bed. it will take a while, but he will eventually.
The funny thing is, that when I am having a bad day, people who know us, will tell me how lovely my ds is and how much they like him. Every time I am amazed. I only remember the public temper. The other people remember the good stuff too.