"satan" is
an accuser. (10 times or so in OT)
"ha-satan" is
the accuser. (12 times or so in OT)
I can't tell you in particular where each might be used in scripture.
Serpent, snake, viper etc., are persona applied to any person, group, or idea, which acts like a snake in it's most simple traits.
A serpent is someone or something that is a "deceiver", because a snake's movement is deceptively forward, while you are distracted by the wavering movement between the "left" and the "right". A deceiver between left and right, two perspectives, good and evil.
Secondly,
if it were to bite, it's bite
may mean pain, sickness, or death.
Thirdly, though not applicable to Eden, scripture makes a further distinction between "viper" and "asp".
The most reputable trait of a viper is that it controls the amount of venom, and therefore the outcome based on it's state of mind and intent. They are also relatively slow.
The most reputable trait of an asp is that it is used
as a tool for inflicting death in high profile death sentences and royal suicide.
If a man was killed by having his spirit crushed by a deceiver, I think it would be accurate to call that deceiver a "python", though I haven't encountered "python" in scripture.
So if you know anybody like that, they are a "snake" and this along with many other coats of skin are preserved throughout history in today's speech across many languages. The languages were confused in Babel, but the meanings of the words remain the same except where graven images have been made.
So to compare the accuser and the snake as if they are both the same in the garden of Eden, might suggest these points:
1.
an accuser, satan, applies to the serpent, because he did accuse God of holding something back from Eve, and of lying about the penalty of death.
2.
the accuser, haSatan, "the boogie man himself" might or might not apply, but I would think so, because he is treated like a familiar particular snake in the dialogue. But it can be equally argued that the word addresses "the idea of the deceiver" when it talks about "the serpent".
3. serpent, "deceiver", certainly applies because reportedly that serpent explicitly deceived, or beguiled Eve.
So the serpent in the garden is certainly "satan" going by the meaning of the word (
an accuser) and his actions, if not "the Satan" himself.