First, I want to give my personal impression. (Keep in mind that I'm just learning Japanese, so I have less experiences to call on than some other people! But I think perhaps I've seen enough Japanese to give an impression worth sharing, nonetheless.)
My impression is that 字引 is just another word for dictionary, but that it's quite a lot less common. The words I see most often are 辞典 and 辞書. In contrast, the only place I can really recall seeing 字引 used is on Wikipedia, in the "Wikipedia is not a dictionary" template:
And notice how it says 辞書 in parentheses after 字引, telling you what it means. I think in this context it pretty much just means dictionary, and I haven't come across it in any other contexts that I can recall.
Well, there is one more place I've seen it, which is the word you mentioned:
生き字引 ([a person who is] a living dictionary). But you knew that one already.
We can also check a dictionary. Kotobank has a single page with the definitions from both 大辞林 and 大辞泉, so I'll link to that. Notice that both dictionaries basically just give a list of synonyms. Between the two dictionaries, they list 字典, 辞典, 辞書, and 字書.
The only thing I could guess is that because it has 字 in the name, it's more likely to be used for dictionaries indexed by characters, such as 漢和辞典・漢字字典. And that's supported by the dictionary definitions which refer to 字典 and 字書. But they also list 辞典 and 辞書 as synonyms, and I haven't actually observed that in practice, so I don't know how true it is.
Is it really that rare? I decided to check 少納言 to find out. Here are the terms I searched for, from most results to fewest:
字引 - 生き字引 ２０件
total - 生き字引 ２１０８件
I included 生き字引 so we could see that it accounts for almost half of the uses of 字引 (17 out of 37). So taking those 17 results out of the equation, it seems that 字引 accounts for less than 1% of the total results (20 out of 2108).
Anyway, this more-or-less fits my intuition that 辞書 and 辞典 are the commonest words for dictionary. It seems that 字引 is a rare alternative which appears in the compound