What does "voice" mean in the context of written language?

Why is the grammatical category of "voice" so named?

Good question. I wondered about that, too, and figured it was Latin -- vox, vocis, after all -- so I checked Donatus. But he doesn't use the term at all, and doesn't even group active and passive in a separate category; they're called 'types of verbs' (genera verborum -- given the definitions, this seems to be like what we would now call "verb forms"). Passive and Active verbs are identified only by their endings, not by their uses or meanings, and they're classed with deponent and semideponent verbs, also by types of endings. So that's a dead end. Grammatical "voice" is not a Latin term.

The OED's first examples of the grammatical sense of voice in English are remarkably recent:

• 1382 Wyclif Prol. 57 A participle of a present tens, either preterit, of actif vois, eithir passif.
• 1591 Percival Span. Dict. C 2 By changing e of the future of the Indicatiue into ia, you make the third voice of the preterimperfect tense of the Subiunctiue.
  (the OED adds that it was "used instead of 'person'" here; i.e, it's a typo)
• 1612 Brinsley Pos. Parts (1615) 20 b, Giue the terminations of the first Persons of the Actiue voice alone.

These appear to be discussing Latin grammar. Looking at related languages, German just says im Passiv, but French and Spanish both use reflexes of Latin vox.

So somewhere along the line, the term 'voice' got picked up and stuck with this meaning. Probly nobody had any use for the term any more, what with all this newfangled printing and reformation and renaissance and all that stuff going on.

English, of course, doesn't have any grammatical Voice; there's a Passive construction (transformation, rule), and a Middle alternation (which, incidentally, is what's going on with The book is selling well), but no Active construction, rule, transformation, or alternation. Or voice.

This is just linguistic terminology, used conservatively. I'm not a conservative person, but I use grammatical terms conservatively because when they're used liberally, they tend to smear across every topic we can possibly associate with language, which is pretty much everything.

So, is there a common thread to these different ways for referring "voice" in written language?

Yes, it all relates back to the idea of a noise produced by a person (or later, an animal or other sentient creature.) The OED gives 3 main meanings of "voice" Together these have 14 sub-meanings and about 36 nuances and uses. It would be unreasonable to reproduce them all.

I. Sound produced by and characteristic of a specific person or animal. (A particularized instance of the phenomenon described in branch II.) From c.1300

II. The sound that can be produced by the vocal organs of humans or animals, considered as a general fact or phenomenon. from c1330

III. Grammar.

A category used in the classification of verb forms serving to indicate the relation of the subject to the action.

c1425 in C. R. Bland Teaching Gram. in Late Medieval Eng. (1991) 160 (MED) Þo secund coniugaciun..of passyf wowus, þat as -e- long befor þo -ris indecatyf, as doceris.

a1450 (▸a1397) Prol. Old Test. in Bible (Wycliffite, L.V.) (Cambr. Mm.2.15) (1850) xv. 57 A participle of a present tens, either preterit, of actif vois, eithir passif.

1991 ‘J. le Carré’ Secret Pilgrim vi. 128 An effort now being made, he said—making suspicious use of the passive voice.

The noun itself comes via Norman French and Latin (vox) from the same Indo-European base as Sanskrit vāc, Avestan vač, ancient Greek ὄπα (accusative).