Letters Testamentary is the correct term. And, wow, Texas still calls a female Executor an Executrix? Very quaint. (I actually prefer it, since it lends an air of mystery and intrigue to an otherwise very dull occupation. I think you should ask for Letters and a whip. Oh, wait, different sort of -trix...)
If there were no Will, or the Will did not name an Executor, or something freaky was going on (someone died mid-administration, the Will named a minor or someone overseas, etc.) you'd have to have someone appointed as Administrator. That person would obtain Letters of Administration of some sort -- e.g., cum testamento annexo (with Will attached), de bonis non [administratis] (goods not administered), durante minore aetate (during the minority). Things can get messy and annoying when you have to have an Administrator appointed, but at least you get a cool Latin title.
Correct
If there were no Will, or the Will did not name an Executor, or something freaky was going on (someone died mid-administration, the Will named a minor or someone overseas, etc.) you'd have to have someone appointed as Administrator. That person would obtain Letters of Administration of some sort -- e.g., cum testamento annexo (with Will attached), de bonis non [administratis] (goods not administered), durante minore aetate (during the minority). Things can get messy and annoying when you have to have an Administrator appointed, but at least you get a cool Latin title.