The two twentieth-century readings are attributable, mutually incompatible, and both routinely presented without names.
Freud placed the snake among the symbols standing for the male genitals, a reading he considered so evident that he spent little effort defending it. It is part of his general position that the manifest content of a dream disguises a latent wish, and that a limited stock of symbols does the disguising. Whether one accepts the theory or not, the reading is his and rests on his argument.
Jung rejected precisely that approach. He argued against fixed symbol dictionaries repeatedly, insisting that the same image means different things for different dreamers and that the interpreter must work by amplification and by attention to the individual's situation. His own reading of the serpent, as an image of the instinctual psyche, of what is coldblooded and other within a person, and of dangerous transformative energy rising from below, is a proposal within that framework, not a lookup table entry.
Kundalini as a coiled serpent at the base of the spine is a real doctrine of Indian tantric physiology, and it is not a theory of dreams. Importing it into a dream dictionary entry, as is now standard, is a category transfer performed without acknowledgement.
The result is that the modern entry, transformation, healing, hidden threat, awakening life force, is a composite assembled from four traditions that disagree with one another, presented as though it were a single ancient consensus. The honest version lists the readings, names the holders, and notes that they cannot all be right.