Learn English – Correct term for “primary” and “secondary” hand

For dominant hand, the only word I can think of which is synonymous in context is preferred.

Right-handedness: The word adroit is defined to mean dexterous, deft or skillful. As the OP has noted, dexter is Latin for "on the right". Digging a little deeper, the etymology of adroit reads thus:

1650s, "dexterous," originally "rightly," from Fr. adroit, from phrase à droit "according to right," from O.Fr. à "to" (see ad-) + droit "right," from L.L. directum "right, justice," accusative of L. directus "straight".

Furthermore, the French word for right-handed is droitier and maladroit ("inept; clumsy; awkward; not adroit"), while rarely used, can also mean left-handed. The more mainstream term for left-handedness in French is not all that much better, as it is gauche ("Awkward or lacking in social graces; bumbling.").

Another option would be hand-dominance or hand-dominancy as a single word. It is not as common as handedness and I usually see it in medical contexts. (It is usually in two words but you can hyphenate to form a hyphenated compound noun).

Research shows that hand dominancy can be an important feature affecting one's tolerance of pain. According to a 2009 Israeli study from the University of Haifa, right-handed people are more tolerant of pain than left-handed individuals.

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It might encompass the left-hand-dominance and right-hand-dominance better than handedness. Handedness can be better understood as a broader term that covers mixed-handedness and ambidexterity also.

They are mentioned as synonyms in some contexts but here is an explanation of the differences between hand dominance, handedness and hand preference:

Hand dominance means that one hand has the most influence or control. Handedness means that one hand is more reliable for use across a range of skillful acts. Hand preference means that one hand is preferred or chosen.

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Additionally, there are eye-dominance and ear-dominance; and there is the hypernym side-dominance that covers all different types of dominance.