The primary meaning of “testament” is in fact a will; a statement of your wishes regarding the disposal of your property after death. It came into English from the Latin testamentum, where it meant precisely that.
“Last will and testament” is one of those duplicated legal phrases which originally made sense. Your will was a document setting out how you wanted your real property disposed of (to the extent that you could dispose of it, in the days when inheritance rules were rather more rigid than they are today); your testament set out how you wanted your personal property disposed of. If you dealt with both real property and personal property in the one document, it was a “will and testament”.
The Greek word διαθήκη could refer to (a) a testamentum, in the Roman law sense, or (b) a solemn covenant between living parties - in Roman law a pactum. The Greek word is used in the Septaguint to refer to the biblical concept of Covenant, and it’s used in the Christian scriptures to refer to, e.g. a “new covenant”. It’s basically due to poor translation from Greek to Latin that the word “Testament” came to be applied to the Hebrew and Christian Scriptures; the translator failed to appreciate that διαθήκη had different Latin equivalents, depending on context.
Lawyers refer to a letter of the kind the OP refers to as a “post mortem letter”. If it deals with some aspect of the adminstration of the deceased’s estate, but is not intended to be legally binding, just to suggest things that the executors might bear in mind when making choices open to them under the will, then it’s a “precatory letter”.