There are no “rights” and “wrongs” when it comes to defining “street photography.” It is your definition. For me, street photography is documenting humanity. It can be done in a public space (subway, mall, park) or can even be done in indoor public spaces.
There are a lot of different styles in street photography (candid, street portrait, witty observations, urban landscapes). Find the style that best fits your voice, and shoot from the heart.