Is alpha thalassemia trait the same thing as alpha thalassemia?

Not quite. Normal people have 4 Alpha globulin genes. Mutations in 1, 2, 3, or all 4 of the genes cause the 4 levels of Alpha thalassemia. One mutation = silent carrier (no symptoms), 2 mutat. = Alpha trait (mild anemia), 3 mutat. = HGB h disease (anemia sometimes needing transfusions), 4 mutat. = bart's disease (fatal before birth in almost all cases; lifelong transfusions needed to survive if born alive).
Thalassemia vs trait. Thalassemia is a deficiency in production of one of the globin chains. There are many variants that manifest a wide severity of symptoms from none to transfusion dependence. In the case of Alpha thal, the term is a catch all for all symptomatic variants of the disorder. Alpha that trait specifically refers to a two gene deletion, in either cis (same chromosome) or trans (different chromosomes).