The terms vary depending on the area you are talking about, and also who is it that is defining them. The Census Bureau defines Cary/Raleigh as urban. But the Census Bureau only distinguishes between urban and rural, so it is not useful for making finer distinctions. The distinction between “urban” and “suburban” is often one of those “I know it when I see it” sort of thing; it is hard to make an objective definition that doesn’t have glaring exceptions. And there’s edge cases, like the many small towns such as Apex and Fuquay Varina that you’ll find with single-family homes on postage-stamp lots. As for the “inner city”; generally it refers to parts of cities with high minority or immigrant population, poverty and crime. If you want to refer to the central area of a city without that implication, then use “downtown” or “central business district”. Cities with multiple area like this will be referred to by a local name.
Now you might not think of Cary as an urban area. You might now even see Raleigh that way, perhaps you see it as a Major City. That is when we go back to the “I know it when I see it” reference. Those who live in the area might define it differently than those who live on the outskirts. Or may have never been there before. What do You think defines the cities in the triangle area as Urban or Suburban?