You’re exactly right, the Rod of Asclepius (one snake, no wings) is the authentic ancient Greek symbol of healing, directly linked to Asclepius and the cult of medicine. It appears in many ancient sanctuaries, inscriptions, and reliefs, especially at Epidauros. The Caduceus (two snakes, wings), by contrast, belongs to Hermes, symbolizing commerce, travel, and mediation.
The confusion really is a modern one. In the U.S., the Army Medical Corps adopted the Caduceus in the early 20th century, partly because it “looked” medical and had classical associations. Over time, that spread into hospitals and pharmacies, even though it was historically inaccurate.
Ancient Greeks did not use the two symbols interchangeably — their meanings were distinct. Archaeological depictions of Asclepius consistently show the single-serpent staff, while Hermes always carries the winged Caduceus. The modern blending reflects more a branding error than an ancient tradition.
Your confusion is because Asclepius [ Eshmun ] and Hermes are one and the same. Apollo [ Aplu Enlil] is Nergal [ Ares ]. Ares is Mars. Mars is Celtic Camulos.
Nudd [Nuada/Nodens] has been left out of the genealogy. Hermes is Arma a Hittite Moon-god. Moon also called Fionn/Finn in Gaelic ,Gwyn/Gwion in Welsh, Leucus in Greek and Laban [Alban in Latin] in Middle East; all meaning white.Nudd was the son of Mars, but at Lydney confused with Mars. Nudd is Nuada

/Nodens linked to Etruscan Nethuns and Roman Neptune [Sydyk in Phoenicia ,Sutekh in Egypt]. See Hermopolis -
Arab Ashmunain or Eshmunain [Eshmun means 8 ,see 8 gods of Hermopolis].
Arsinoe was the mother of Asclepius and the daughter of Ares and Ino [Inanna/Aphrodite/Coronis/Periclymene].
Ophiuchus is the Babylonian god Nirah [Nireus/
Orestes in Greek mythology]. Aglaea is ugly Medusa [Hittite Hannahanna is Diana;Hannahanna means grandmother].
Grandmothers or aunts [Tante/Tanit] act as nannies to their grandchildren. Charops is an epithet of Heracles [Poseidon/Palaemon].
Manawydan [Manannan/Mani/Men/Meness] was the son of Ma/Maia and Nuadan.
The myth where Heracles fights Apollo for the tripod at Delphi relates to the three-legged depiction of Pegasus [Sleipnir/
Asleep/Asclepius/Hypnos/Morpheus/Erebos/Horeb/Oreb/raven in Hebrew/Sinai/Sin/Corax/Ceryx/Odin/
Hermes] in the stars. Pegasus is depicted in the stars as a seahorse, the mount of Poseidon [Nodens/Nethuns/Neptune/
Nuada/Nudd/Ea-Nudimmud/Endymion of Heraclea or Elis and father of Epeius/Norse Nidung].
Asclepius' wife is Epione [Epona/Norse Hildr/English Hilda/Athena Nike/Frigg/
Jord/Mother Earth/Gaia/Semele/Zemelo/
Zemlya].
Wayland the Smith [Hephaestus/Lycus] slew the sons of Nidung [the 8 sons of Heracles].Eshmun means 8 in Punic/Phoenician, but closely resembles
Latin Somnia [Hypnos] and Irish Samhain.
At Feis na Samhain [1 November] the dead were supposed to rise [as ghosts].
Melampus is a name of Asclepius [Hermes].
In Ireland, Bodb Derg was Welsh Bran [Budha/Brahma/Mercury/Thjazi/Adam], the brother of Aengus Og [Anguis og/Ophiuchus/Asclepius/
Manannan/Odin/Odysseus/Hermes].
Lir is Etruscan Laran [Mars]. Aoife is Antiope, the mother of Zethus [Set] and Amphion [Osiris/Aed].Fionnuala is Selene.