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

If so much of CA's water wasn't being sold in bottles as far away as where I am, there might not be a need for such restrictions.