So I turned to the OED.
I. Power to enforce obedience.
All the entries under I. are going to be problematic, because the Bible cannot have “power” in this sense; it is a text and not an agent. But let’s see in detail.
I.1.a. Power or right to enforce obedience; moral or legal supremacy; the right to command, or give an ultimate decision.
Now this one seems appealing, but it can’t be right, because the Bible does not have the ability to enforce anything, and it demonstrably does not give an “ultimate decision”: that task gets referred to interpreters.
I.1.b is the definition of “in authority,” not relevant here.
I.2.a Derived or delegated power; conferred right or title; authorization
Perhaps the Bible has had God’s authority to it; it speaks in the name of God. That sounds good, happily close to the sense of the the Bible as the Word of God. But, again, alas, it can’t be right, because again, this is about power and agency.
I.2.b is the definition of having authority to do something.
I.3 Those in authority; the body or persons exercising power or command
This is a metanymy from since I.a and I.b; the word shifts from the power itself to the one exercising power. (This is the difference between saying that the Bible has authority and is an authority; but the fundamental problem with all of these in I. still remains.)
II. Power to influence action, opinion, belief
This seems more hopeful, but notice that the difference between I. and II. is the difference between compelling obedience and merely influencing. The question of how this is supposed to work: of what exactly this power is supposed to consist in, in the case of a text, is still unanswered. So we expect with the II. definitions there to be problems similar to the I. definitions. But let’s see in detail.
II.4 Power to influence the conduct and actions of others; personal or practical influence
Here again, authority is necessarily vested in a person or group of persons, again, because the notion of agency is central.
II.5 Power over, or title to infuence, the opinions of others; authoritative opinion; weight of judgement or opinion, intellectual influence
Ah, this sounds good! Again, however, this authority is the authority held by authors, writers, and such, it is not an authority of texts. We might say, in this sense, that Paul has authority (heck, he might well have authority in sense I.1.a), but not that his words do. And it might well be impossible for a dead person to have authority in sense I.1.a, but a dead person could have authority in sense II.5. But still, not so much the text.
II.6 Power to inspire belief, title to be believed; authoritative statement; weight of testimony.
Ah, this is getting there. Here indeed the example quotations in the OED have some relevance, referring for the first time to texts as well as persons.
II.7 The quotation or book acknowledged, or alleged, to settle a question of opinion or give conclusive testimony
And this is clearly even closer to the meaning; by saying that the Bible is authoritative, one says, perhaps that it is or ought to be a source which is used to settle a question of opinion or give conclusive testimony.
II.8.a The person whose opinion or testimony is accepted; the author of an accepted statement
No, this is clearly not it, nor is:
II.8.b One whose opinion on or upon a subject is entitled to be accepted; an expert in any question
II.9 is the definition of “authority-maker.”
So there you have it. It seems to me the sense must be something like either II.7 or II.6.
Now there is a problem. If the Bible is authoritative in sense II.7, we are stuck, because it isn’t working. In other words, the Bible could fail to be an authority in sense II.7 not because it is in error, but because looking at it doesn’t actually settle the question of opinion which vexes a given group. For example, in Scrabble the agreed dictionary is the authority (sense II.7) for disputes about legitimate plays. But suppose you opened the dictionary, and found that it was written in a script that none of the players could make out clearly, and you immediately have a dispute about whether that black smudge is an E or an F. A text can therefore fail to be an authority in sense II.7 simply by being unsuited to the task of settling disputes. The official edition of Kant’s works is an excellent authority for his words; it is not an adequate authority (sense II.7) for his ideas, because Kant scholars, of course, disagree wildly about what he meant by the various sentences on the page.
At the Reformation, however, the propaganda from the reformers, and sadly still from so many of their heirs, is that the Bible is an authority in sense II.7. The Anglican formularies, and even more so the Calvinist ones, speak of this kind of authoritative role for Scripture. The pope can, of course, have authority in sense I.1.a, but a text cannot; the text can, at most, settle disputes in the sense of II.7, but this requires it to actually work. (The pope can be authority in sense I.1.a even if he never speaks and uses that authority; but the authority of II.7, the only one of this kind a text could have, it can only have if it, in fact, speaks effectively to its hearers, and to the task of settling the dispute.)
So what about II.6? That’s ok, but notice that it doesn’t license the kinds of conclusions people seem to want. The Bible can have that kind of authority and be utterly unable to settle any disputes. You can’t say that someone doesn’t give it authority merely because you disagree with them about what it means.