Sedges generally do not have rounded stems - they will be flattened or triangular in shape and solid....grasses are rounded and usually with hollow stems (although at that size, that may be hard to determine). Sedges also tend not have any sheaths (the dry material showing). In any case, if it is not something you want in your lawn, it is safe to consider it a weed :-) RoundUp/glyphosate will take care of it but it may need several applications before full control. And it will also kill off any of the desired turf grass it comes in contact with. It is also getting pretty late in the season to be overseeding in your area now. Tricky to get the new grass to grow and develop a decent root system before really cold weather sets in. You might want to repost this in theLawn Care forum. There are a couple of lawn gurus who frequently post there that are very good at ID'ing weed grasses and recommending specific herbicides that will address these and not kill off the rest of the lawn as well. And they can also explain the ins and outs of reseeding and when best to do so in your area. But I am afraid you might have missed the mark this season.

Yes. It does look like Star of Bethlehem, and it does have tiny bulblettes. I looked it up on Google. Can you believe this plant is sold online?

Those are terrible photos - almost impossible to distinguish what the weeds look like in that indistinct pulled clump. Next time take a careful shot of them growing in place. As best as I can tell, that looks like ground ivy, Glechoma hederacea. That is an evergreen creeping vine that will invade lawns and flower beds. And any pre-emergent will have no effect on it. Even a very lush, heathy lawn will have difficulty out competing ground ivy. Typically it will require the use of post emergence lawn herbicides containing dicamba, 2, 4-d and triclopyr (like Trimec). Fertilizing the law will only serve to fertilize the ground ivy as well.

It looks like clumps of warm season grass growing in a cool season lawn