Your lawn weeds are both broad-leaved plants that grow best in moist or wet conditions, spread by rooting stems and also by seed, meaning they expand quickly and are difficult to remove by hand.
You'll find herbicides for sale listing these plants. Use as the label instructs. There are likely many seeds in the soil, so establishing healthy turf to limit weed seed sprouting will be important.
The first plant is one of the self heal (Prunella vulgaris) weeds. Here is an information page from UC IPM: http://ipm.ucanr.edu/PMG/WEEDS/healall.html
The second photo shows one of the Polygonum, often called smartweed or knotweed. Here is UC IPM about one of those, ladysthumb: http://ipm.ucanr.edu/PMG/WEEDS/ladysthumb.html
Establish a lush lawn by renovating or planting in the early fall is an article from Oregon State University which you may find helpful: http://extension.oregonstate.edu/gardening/2016/09/establish-lush-lawn-renovating-or-planting-early-fall
Finally, I recommend you contact your King County Master Gardener Volunteers for local turf care suggestions and current rules on herbicides. http://extension.wsu.edu/king/gardening/mg/