It was indeed the original snake God made, or one of the kinds of snakes He originally made.
Baraminologists believe there are 3 basic kinds of snakes which all species are derived from. At least some believe this.
Most creationists believe the snake in the garden originally had legs, which God removed after the Fall. It's likely God altered many animals after the Fall, indicated by the phrase "above all livestock."
Gen. 3:14 So the LORD God said to the serpent, “Because you have done this,
“Cursed are you above all livestock
and all wild animals!
You will crawl on your belly
and you will eat dust
all the days of your life.
The implication is that it was altered after the Curse along with other animals. This is in line with Jewish tradition. Josephus believe God took the Serpents speech away and put venom under his tongue.
Antiq. 1:50 (1.1.4) He also deprived the serpent of speech, out of indignation at his malicious disposition towards Adam. Besides this, he inserted poison under his tongue, and made him an enemy to man; and suggested to them that they should direct their strokes against his head, that being the place wherein lay his mischievous designs towards men, and it being easiest to take vengeance on him that way: and when he had deprived him of the use of his feet, and made him go rolling all along, and dragging himself upon the ground.
Josephus believed other animals had speech before the Fall.
Antiq. 1:41 (1.1.4) But while all the living creatures had one language,5 at that time the serpent, which then lived together with Adam and his wife, showed an envious disposition, at his supposal of their living happily, and in obedience to the commands of God;
This all happened before the first animal death, so none of these traits were passed on to descendants.