Richard:

in normal English under current conditions because Texas is part of the U.S., a native Texan by definition is natively an American (a citizen of the US).

However, the term "Native American" got borrowed by the politically correct to apply to those who lived in the land which is now the U.S. before the United States existed. These "Native Americans" ar ewhat used to be called Indians. Their descendants of the various tribes are referred to as Native American by birth.

Texas was originally part of Mexico, and for a brief period became its own Republic before joining the US.

All clear?