"central city : a city that constitutes the densely populated center of a metropolitan area — called also core city." -- Merriam-Webster
"suburb: a smaller community adjacent to...a [central] city." -- Merriam-Webster
Example: Atlanta and its suburbs
"Metropolitan Statistical Areas (MSA) have at least one urbanized area of 50,000 or more population, plus adjacent territory that has a high degree of social and economic integration with the core as measured by commuting ties." -- US Census Bureau
"Combined Statistical Areas (CSA) consist of two or more [metro areas]." -- US Census Bureau
Example: Stanford University is in the San Jose-Sunnyvale-Santa Clara MSA and in the San Jose-San Francisco-Oakland CSA.
2020 State-based Metropolitan and Micropolitan Statistical Areas Maps