I noticed that usually if there is an area between streets, it’s not connected to the street nodes, but drawn as a separate polygon. What if the area is directly adjacent to the street? I see that I can connect the nodes directly to the street, and it even will show up correctly in the map, but the portion of the street that’s sharing the nodes with the park now will have incorrect tags, so I gather that this is not the correct way of adding areas.
I haven’t seen any specific guide about this, everything talks about nodes and ways, but not about closed ways ie. areas
The portion of the street that is sharing nodes with an area is not having incorrect tags applied. The street and the area will still have their own way, with respective tags.
I always leave a small gap as well. My reason being that the way for the road is describing the centerline of the road, and an area usually does not start in the middle of a road. It also makes for easier editing, but that is not my main reason.
The problem in your example is that the park is tagged incorrectly: http://www.openstreetmap.org/browse/way/30195704
The boundary of the park area shouldn’t be tagged highway=tertiary. Delete that tag and the extra name rendering will disappear (as there is no name tag currently on the actual tertiary road).