Well, the new house has a mystery tree. Our initial guess was some form of a broad leaved ligustrum. It appears to be evergreen - no leaves have been lost yet although the neighboring peaches have lost all their leaves in the past week, and the post oaks have lost almost all their leaves.
Height is a little over 12 ft. The leaves almost remind me of an apple - but they're glossy/waxy (unlike an apple).
The bushes in the background are small-leaved ligustrum ("privet" in the UK). Ligustrum is an invasive here in Texas — the bushes will be replaced in the next 6-12 months. The identification of this tree could seal its fate.
Spring Addenda: Flowers are white with five petals. They did not last long and unfortunately I did not get any photos. More like blossom than a ligustrum flower.