The weed in your first photo looks like false/mock strawberry. Did you notice if the flowers were yellow? Yellow= weed, while a white blossom would be a native wild strawberry, which are much less common. The weedy one produces small red berries with tons of seeds on the outside as you will see in the photos on this page from our website: 

https://extension.umd.edu/resource/indian-mock-strawberry

Your second photo shows a type of Smartweed. Here is our page on that: https://extension.umd.edu/resource/smartweed  Smartweed is an annual which can seed itself each year when allowed to flower and mature. It's not usually a big problem in lawns, but we'd suggest pulling this up and putting it in the trash since flowers are evident to prevent it from going to seed.

Similarly, if you just have a small amount of the Indian Mock Strawberry, just pull each plantlet up and dispose of them.

Christine