Could be Ranunculus or something in the waterleaf family based on the pic in your other thread.

In your photo there are LOTS of little seedlings, which means there must have been lots of seeds in your lawn or in the compost you put down. You didn't mention the lawn having lots of weeds previously so I wonder about the compost. Where did it come from? Your options are limited while the new grass is just getting started since many broadleaf herbicides can damage the grass seedlings too. If you wait until the grass is established at least 8 weeks, then you may be able to use herbicides. Check the label of any product before using it on new lawns. I agree with kimmsr that correcting soil problems is the way to manage moss in a lawn. Moss is not a competitive plant, as you can see by the very limited root system it has. It grows in lawns where the grass is already doing poorly. This can be cause by lack of soil fertility (the most common cause in my experience), too much shade, heavy soils that drain poorly, and excess acidity (pH below about 5.2). Lack of organic matter isn't normally limiting for grass, but adding organic matter can improve drainage if that is the problem. Have you had a soil test done? You mention that the soil was determined to be acidic. What was the pH?

Gnaphalium palustre, lowland or western cudweed. It should respond to any broadleaved lawn weed herbicide but it IS an indicator plant for wet soils.......which might be a concern for good lawn establishment.

I think you might be safe ignoring our typical warnings for summer applications of anything. How often do you get temperatures above 85 in the summer? If your temps typically run in the high 70s to low 80s, then you can do what you want. You have what the rest of us would consider to be fall type weather while we're suffering in more heat. The white flowered plant is Dutch white clover. The yellow flowered plant is black medic. Between the two of these weeds, I would much rather live with the clover. At least it becomes dense and looks fairly lush. Black medic just looks scrappy. To eliminate these completely requires something like Weed-b-Gon Chickweed, Clover, and Oxalis Killer. If you think you can get rid of these organically then you'll have a future of internal conflict with reality. The mix you are using with vinegar will kill the tops of many plants, but only rarely kills the root. Plus it works best on hot sunny days...which you don't really get up there on the lake. I am a 100% organic fertilizer kind of guy and have been for 15 years. At the same time I see the value to having a nice lawn with no weeds. What you might consider is using the chemicals one or two times up front to get rid of them, then follow that up with a grass "densification" program to help keep weeds out in the future. Reseed the thin areas with Kentucky bluegrass. KBG is the kind of grass that will spread to fill in thin areas in the future. In your area it may go through the summer with only rainfall. Otherwise, it may go dormant in the summer dry spell and/or heat, such as it is in Racine. Fall is generally the time of year to reseed northern grasses. In your case you might start toward the end of July and get good results. My organic fertilizer of choice is alfalfa pellets (rabbit chow). I like it because it is available at every feed store in my area, and it works. You might find that something like soybean meal, corn gluten meal, corn meal, cottonseed meal, or Milorganite is more available to you. Certainly Milorganite is available in Racine!! The application rate for all of these is 15-20 pounds per 1,000 square feet. You might be able to get a discount on a pallet for your acreage. If you live in an affluent area and want to keep your lawn up to their standards, it's going to cost you some money.