It's fluffy but not appealing or easy to mow

I don't know about your area, but its green. Where I live that is an important criteria on whether it is "grass" or not

Oh it's not grass, it's leafy. Southern Maryland is the area

I could not name it but, here's about all the research you need: It's a broadleaf plant so use a broadleaf plant killer like Weed-b-Gon or Weed-b-Gon Chickweed, Clover, and Oxalis Killer. These WBG products will not hurt your lawn, but you will need to reseed in a few weeks. Don't put off reseeding until spring. Do it early in the fall so you have time to evaluate it and rereseed any thin spots.

I see a lot of buttonweed and clover, which I usually associate with shade.

Well now we know what it's called, it is buttonweed. I had great success at the beginning of the summer with 2,4D on it but now I have a few plants which have gotten a little bigger and they seem to not respond as well to it. Going to try triclopyr next.