I think of exurbs as areas that aren't connected to the city via a solid strip of urban land, but still are heavily economically tied to the city. I think this is more common in the West where there are a lot of reasonably dense urban areas with virtually empty land between them. Suburbs are just parts of an urban area that aren't politically part of the core city of that area for whatever reason (although in my experienced it tends to refer to smaller and often wealthier examples of those communities).
"Outskirts" I hear more in reference to small towns, and it refers to the rural area immediately outside of the town. If someone told me they were from the "outskirts of Boston" I wouldn't really know what they meant.
Interestingly, if someone told me they lived on the outskirts of Boston I would assume a town near I-495 without a second thought. I went to school in the suburbs of Boston (ie near rt 128), but my in-laws are from just inside the outer ring of I-495. I've even heard them refer to their towns as on the outskirts of Boston.
Boston itself is a small city - with places like Cambridge, Allston and so forth identifying closely with Boston.
Then you have the places inside the 128 ring that are strongly affiliated with Boston and long had both commuter rail and metro stops. Places like Newton, Milton, and so on. Further out on 495 you have places like Franklin, Ashland, Mansfield and so forth that have a commuter rail station but are typically a leafy, suburban 'bedtown' which some people say outskirts when referring to Boston.
I grew up in North Attleboro and we seemed to have more affinity with Providence, RI than Boston though some of our neighboring towns closer to 495 - like Foxboro and Mansfield were closer with Beantown just a few miles away to the edges of these two towns. In that we bordered RI and I-295 and that it took less than 20 minutes to get to Downtown Prov and almost 40 on a good day to Downtown Boston.
In SE Michigan for exurbs I would say a place like Howell, MI - where it is a good 30-40 minutes from the more densely settled suburbs where the office park type jobs are and a good 1 hr plus on most days to downtown and people do things like ride share. It is definitely a small town / city area that has some separation from the metro area but due to all the people working and visiting there it is very connected with it. These are not as common in the Northeast's Megalopolis between Metro Boston and Metro DC but are quite common in much of the rest of the country. The terms is more for academic purposes and in writing and not used in common spoken language.
