Street photography, when boiled down to its essence, is simply candid photos of people in a public space. There is plenty of historical precedence for this definition, and anyone who is considered a famous street photographer will have all of their famous street photographs fit this description. I think where the controversy begins is when you start straying from this definition, particularly if redefine it by removing words. Take out candid, people, or public, and suddenly great debate will emerge. Personally I think the core issue is there aren't any well defined labels for these variations. Some terms that get used include street portraiture, urban landscapes, and candid portraits respectively. But if you took a poll you will likely get as many responses as respondents. I have no particularly strong feelings on how to name things, but I think it's useful to have generally agreed upon labels.
The same things that make any photograph good. Light, composition, subject. What makes street so incredibly challenging is how little control of these things you have in the moment. And moment is the key word, whether you want to append 'decisive' or not (I personally think it is a nonsensical distinction, as if one moment is somehow better than others). The moment you see something interesting you have two choices; take the photo from where you are as quickly as you can, or start scanning to see if you can move to change the background, wait until the subject enters better lighting, or simply reposition yourself or wait for the composition to come together. The sheer number of almost good images you will get from not being able to control these elements is one of the two most difficult aspects of street photography.
Legally, in the US, you have every right to take a picture of anything you can see while you are standing in a public space. Morally you are certainly within your rights to be taking pictures of people who have entered into a public space and are taking no issue with the gaze of strangers, being in the background of selfies, and recorded by any number of surveilance cameras. It is only when confronted with a stranger pointing a camera at them that people tend to panic a bit.
Honestly, so do I, I completely understand where people are coming from. It is disconcerting, and potentially upsetting. For this reason I personally do my best to smile, say thank you and converse with anyone who intiates a conversation with me. I may delete a photo at their request, but 99% of the time I just tell them why I took their picture. There is always a reason, if you are honest with yourself, and I think it speaks volumes to your character as to what that is. "You are wearing a really cool hat" will always, in my experience, immediately ease their concern and more often than not get a smile or laugh. If you pointed a camera at a person in distress, or whom you have much greater social standing and are leering or laughing at their predicament, then you should be willing to tell them as much. And if you get punched in the nose... you deserved it.
Street photography comes from the lineage of documentary photography and photo journalism. Street photos are not generally planned, concepted or executed in the same way a portrait or still life might be. However, there are many, many beautiful, arresting, incredible street photos. And if you can tear a urinal off a wall, tape a banana next to it and print off someone's instagram post and change the art world forever, I don't see why a street photo wouldn't be in the conversation. Personally I have tried to move my street photography further and further in to a style and aesthetic that I want to convey, but obviously much of the work remains serendipitous.
Get low, get close, shoot wide, use a flash. I have tried many techniques to add some interest, mood and simply to personalize my aesthetic. As time goes on I have tended somewhat towards the surreal.