Where in the United States are you? Delaware? Eastern Washington? New Mexico? Florida?
A "weed" is a plant that you don't want growing where you don't want it to grow, so grasses can be "weeds" just as any other plant could be. One big problem with spraying stuff that will kill grasses is that it kills all grasses, even crabgrass killer will kill Quack Grass.
Annual Ryegrass is a cool weather grass that is growing now and that may be what you have. Many people consider this a "weed" while others seem to think it is a good green manure crop. It can be difficult to control but it is an annual and if you prevent the seeds it produces that will help control it.

It does look like ryegrass that has been bird-seeded into a permanent lawn of (probably) bermuda or centipede. In much of the south, overseeding the dormant warm season grass with annual or perennial rye for a green lawn all winter has become all too common. Birds love the seed and are happy to spread it everywhere.

Put your herbicide away. You can just pull the grass or wait until the hot weather comes to kill it.

I saw my neighbor sprayed some green herbicide to kill the spot ryegrass (assume it is as you identified). See the picture

The herbicide probably wasn't green, but was dyed that color so that the applicator could see where he or she had sprayed. Blue or green dye is commonplace among professional applicators to track liquid chemical and fertilizer.

Assuming it is the ryegrass as you identified. Is it Round-up Weed & Grass, Ortho Weed B Gon or anything else effectively to kill it?

It looks like Allium to me, perhaps A. vineale, wild garlic. Pull a stem and smell it, you'll know for sure if it is. I would suggest digging it out as you find it until it goes dormant in late spring.