In these accounts, the mind body problem arises because the mind seems to be so fundamentally different from any physical substance, yet it is obviously correlated with the body.
This is the main problem. The question is how certain physical properties - say, certain brain cells firing in a particular order - can give rise to certain phenomenal properties, such as seeing, tasting, etc. Here's a relevant quote from Chalmers:
The impressive progress of the physical and cognitive sciences has not shed significant light on the question of how and why cognitive functioning is accompanied by conscious experience. (The Conscious Mind, p. 25)
A pithier way I've seen him put this is as "why should this (physical property) feel like that (phenomenal property)." Note that there's a problem only if there are such things as phenomenal properties, and they coincide with physical properties in the way that people like Chalmers think they do. You can avoid this problem altogether by denying that there are phenomenal properties. Eliminitavists like Daniel Dennett take this approach, as do the Churchlands. Most people think that this is a non-starter, however.
With respect to your main question, you note that the dualist has to account for how the mind interacts with the body. This is correct. But it isn't the core of the mind-body problem (if it exists, assuming we're not eliminitavists). This is less of a problem for the materialist. Note, however, that your second question applies to both the dualist and the materialist alike - their positions are distinguished by how they answer the question of the relationship between the mind and the body.