Hello mudsnake,
The plant with purple flowers is Agalinis, most likely A.fasciculata
The first one looks very familiar...

Your first one definitely belongs to Commelinaceae, and I think it might be Gibasis sp.

Here is a link that might be useful: Dave's Garden

Thanks for the ID, Tyrlych!

Both "weeds" are nice looking plants. I had no idea that the Gibasis is a plant normally cultivated.

They are only a weed if you do not want them in your lawn.

Mine is in an OK area far from turf grass.

I suppose that the seeds were dropped by a bird.