"Stable" is actually the game term, "stabilize" is the verb form.
It might be useful to look at the rules for death and dying here:
Stabilizing A Creature
The best way to save a creature with 0 hit points is to heal it. If healing is unavailable, the creature can at least be stabilized so that it isn't killed by a failed death saving throw.
You can use your action to administer first aid to an unconscious creature and attempt to stabilize it, which requires a successful DC 10 Wisdom (Medicine) check.
A stable creature doesn't make death saving throws, even though it has 0 hit points, but it does remain unconscious. The creature stops being stable, and must start making death saving throws again, if it takes any damage...
And just above that, with regard to Death Saving Throws:
On your third success, you become stable...
As the bolded text (which is present in the original) shows, 'stable' is the game term, and 'stabilize' is the verb used to describe becoming stable, but they're used effectively interchangeably in the actual rules that relate to death and dying.