We can't tell from the photos what weedy grass was present, although it doesn't quite look like dormant Bermudagrass. As a perennial grassy weed, Bermudagrass is very hard to eradicate without resorting to completely digging it out and/or treating it with a systemic herbicide like glyphosate. That latter approach will have to wait until it resumes growth for the year, as the herbicide can't work if live foliage isn't present to absorb it. Either approach may cause some collateral damage to the lawn, so you may need to patch those areas with seed or sod (sod will be much easier to establish this time of year, and you can cut it to size if needed). If it's not Bermudagrass, a similar-looking native grass that is sometimes considered a lawn weed is Nimblewill; it spreads in a similar way, and the options for its removal are the same.

The kind of "vinegar" used as an herbicide (we don't know if you're referring to an acetic acid herbicide product or household vinegar, which should not be used on weeds) is very strong and hazardous regarding eye, lung, or skin exposure risks, but it doesn't kill weed roots; it only can temporarily "burn" top growth and force weeds to regrow. That might suffice for annual weeds or seedling weeds with less root energy stored for regrowth, but it's not going to do much to perennial weeds. If the top growth of a perennial weed is repeatedly removed (either manually, like with a weed-whacker, or chemically, with a "burn-down" type of herbicide like that), that can slowly weaken and starve the plant, but for aggressive weeds, that process might take several years and any regrowth needs to be promptly removed or else the weed will have more opportunity to regrow each time it's lost foliage.

Miri