Remove weak and crossing branches and twigs. Keep root suckers removed at all times. In summer, remove spent blooms to encourage more flowers. The plant you mailed is henbit, a cool-season annual that's often called a weed. But, as you note, it is pretty, and I don't consider it offensive or invasive.
Henbit sprouts in the fall and has square stems and pink-purple flowers.
The leaves are opposite, and the margins have rounded teeth. The lower leaves occur on petioles, while the upper leaves are without petioles. The flowers appear in whorls in the upper leaves. You can easily pull henbit. Some use a pre-emergent herbicide for broadleaf weeds in early fall to discourage unwanted plants.