Your understanding of the letters engraved on the stones of the Breastplate (חשן משפט) worn by the Kohen Gadol, like is described by Rabbeinu Bechya to Tetzaveh 28:15, is correct. See also Shemot 28:15-21 for the full description.
Like the Ari z"l, Ramchal and the Ba'al Shem Tov teach, there are 4 expanded forms (called Miloy מלוי) of G-d's 4 letter name (יהוה), which correspond to the descent from higher (greater revelation of spiritual existence and less material existence) to lower (greater material existence and less spiritual). These four general levels correspond to the four general worlds or universes (States of Existence) described as Atzilut (Emanation), Beriah (Creation), Yetzirah (Formation and Asiah (Action).
That name that you mention in your question with a plain Gematria of 72 (יו׳ד, ה׳י, וי׳ו, ה׳י) corresponds to the world of Atzilut.
The "72 letter" name of G-d is a misnomer. The reference to the 72 letter name of G-d by Rabbeinu Bachya is not a separate name. It is because the triplets, like the letters in מלוי, are sometimes referred to as אותוון. This is in contrast to the term אותיות which is used only in the context of single letters.
This is discussed in detail in Pardes Rimonim by Rabbi Moshe Cordevero and Sefer Kehillat Yaacov by Rabbi Yaacov Tzvi Yolles of Dinov.
It is actually a 216 letter name, comprised of 72 triplets (3 letter roots) which corresponds to the phrase, "ריח ניחח אשה ליהוה". See BeMidbar 28:1-8 and specifically 28:7)
That just like smoke rises from the offering burned on the altar, by us (ריח) elevating the גבורה (216) (Severity), we precipitate ניחח, (as in the expression ניחות דרגה or נחת רוח) from G-d, a corresponding downward and opposite flow of blessing into Creation, an effulgence of (72) חסד (Kindness). That transforming the transitory state of Gevurah (216) results in a threefold increase of Kindness (72). That the concept of three also relates to permanence (חזקה).
In kabbalistic nomenclature this principle is called: אתערותא דלתתא אתערותא דלעילא.
This relationship is an explicit teaching from Moshe Rabbeinu coming from the proper alignment of the 3 posukim from parshat BeShallach and the splitting of the Reed Sea in Shemot 14:19-21. The first and third posukim are in correct reading order, from left to right, while the middle posuk is reversed to read from right to left. This principle of conversion from one state of being to another is also related to the first sign that Moshe and Aharon made before Pharaoh in Egypt. This was the conversion of the staff to a serpent and back again onto a staff. This flipping principle also corresponds to the triple blessing recited by the Kohanim in Birkat Kohanim.
This name is what was written on klaf and inserted into the Choshen HaMishpat. The non-Jewish source you mention likely misunderstood the meaning of the Hebrew. That the letters of the name of 72 triplets (actually 216 individual letters) are also referred to as stones (אבנים), because like it says in Sefer Yetzirah, Chapter 1, the letters of the Aleph-Beit are also called stones. And the Hebrew word אבנים means, minimally, two stones. They associated the expression את-האורים ואת התמים to mean literally two stones possibly coated with gold. See Rashi and Ramban there (Shemot 28:30) for clarification.
The Kohen Gadol, when seeking Divine guidance from G-d over a national issue for the Jewish people, meditated on this name while consulting the Urim v'Tumim which caused the engraved letters on the stones to illuminate an answer. The Kohen Gadol would then, via Divine Inspiration (רוח הקודש), understand what message was intended from the illuminated letters, meaning their proper arrangement, word breaks and interpretation.