As far as I can tell, the D&D term "primal" for the Primal power source is pretty good to cover animistic and shamanic concepts within D&D.
Specifically, Primal is the Material Plane, especially the Elementals and Lifeforms (Plant, Beast, etcetera) that are part of the Material Plane.
Importantly, these Material Elementals are specific features of nature, including landscapes, prominent rock formations, waterscapes, prominent rivers, streams, seas, oceans, skyscapes, weather patterns, luminaries, sun, moon, stars, etcetera. All of these natural features are Elemental "persons" expressing desire and a will to be the way they are. Water "wants" to flow, and so on. The features of nature have minds. Similar to how a Human can magically "project" their "soul" outofbody, these Elemental features can as well. Even when the soul projects in this way, the projection is only partial, and much of the soul remains in the original feature of nature. I view animistic "nature beings", such as the Giant creature type as the manifestation of outofbody soul projections of specific mountains or features of a specific mountain. Importantly, these Material Elementals are native to the Material Plane and have little or no connection to the Elemental planes. The manifestation of an outofbody soul might appear in any form, including a humanlike one or animallike one. The soul of a mountain might manifest as dreams and visions of the mountain, or strange aparitions of its features.
Note, the term Humanoid is an umbrella term that means comparable to a Human, socially and cognitively, and with roughly the same potential powers as a Human, such as level advancement from learning and adapting. The Human species is a Humanoid Beast, the Dragonborn is a Humanoid Dragon, the Tiefling is a Humanoid Fiend, etcetera.
The shamanic characteristics translate into D&D as "Primal" magic. By analogy of Psion or Psionicist to the Psionic power source, a shamanic character class might be called a Prime or Primalist to Primal power source.
Perhaps the Primal and Psion power sources should always be "Innate" magic, meaning spells and other magical features never refer to spell components. That said, using voice or gestures spontanously might help to focus a specific magical intention, but there is no exacting magical "forumula" nor specific ingredient. Primal is neither an Arcane protoscientific spell nor a culturally prescribed Divine ritual contemplation. The Primal mage is actually being the qualities of a specific mountain itself, or a specific type of animal, or an other person in intimate communal relationships with them. It is possible for the soul of one kind of nature being to immigrate to an other kind of nature being. The soul of an Elemental mountain might manifest as a Human, truly and fully, while abandoning the link to the mountain. And viceversa, the soul of a Human can become the soul of an Elemental mountain. Notice the Human species is also a kind of "nature being", and is part of the Primal world. Primal magic tends to be "sacred" but it strictly refers to the features of the material world, such as wind-breath and dust-body, and its highest ethic is to figure out how to coexist constructively together.
Note, the D&D Druid includes the option of either Primal or Divine, or perhaps a blend of both. The Primal druidry sacred traditions are known as the "Old Faith" in communion with Elemental and Beast nature beings. The "New Faith" is the sacred traditions that involve cultural symbolism and linguistics that communes with the ideological mindscapes of the Astral Plane.
Primal magic includes the "Border Ether" that is part of the Material Plane, as well as the Positive Energy "Fey Ether" and Negative Void "Shadow Ether". When nature beings project their souls outofbody, the souls move thru the Ether, Feywild and Shadowfell, but they remain within the Material Plane, namely the "Border". Rarely do nature beings venture into the Deep Ether, Deep Fey, or Deep Shadow. Because of the Positivity and Negativity respectively, there are natural "Crossings" from one plane into the next. Thus are Positive Routes of known locations from the Material Plane › Fey Crossing › Border Fey › Deep Fey › Celestial Dominions › Astral Plane. Likewise the Negative Routes connect from the Material Plane › Shadow Crossing › Border Shadow › Deep Shadow › Fiend Dominions › Astral Plane. Likewise, there are Ether Routes connecting the Material Plane › Ether Crossing › Border Ether › Deep Ether › Elemental Chaos › and the varous Elemental Planes that are more like stable regions within the Elemental Chaos. Even a character without magic can planeshift if they happen to know where these location points are and are willing to make the treks within each plane.