I'm looking for the correct term for a right-handed person's right hand, and a left-handed person's left hand, and same for their other hand, where one term signifies one's "primary" hand, and the other one's "secondary" hand.
The "primary" hand is generally called dominant.
Operant hand generally used for performing fine motor-skills tasks (e.g., writing, holding dental instruments)
Medical Dictionary for the Dental Professions © Farlex 2012 via TFD
From that, I would surmise subordinate hand for the other one.
The non-dominant hand is commonly called the off hand. Also the weak hand. See any combat arts forum.