Are These Plants Or Weeds?

When I bought my house I was happy for some of the xeriscaping. I was more than happy to dump those red wood chips all over every year to keep down the weeds and keep in some moisture from the few plants I do have. Since we've been having rainy weather, I've been overrun with these plants.

I guess one mans ground cover is another mans weed. These green leafy plants occasionally pop out small blue flowers and are impossible to control.  All of my gardens are overrun with them.

So is it a weed, or is it a plant? Does it really matter? Well I kind of don't know what to do. Do I pull them all out every year, or be glad that they grow like crazy? I guess you can see that they already ate my Pink Flamingo down to a skeleton.  Any advice from you home gardeners out there?