Here follows some my brief remarks about your question, based on the two Bible passages of Lev 21:7 and Deu 24:1.
On Lev 21:7 we find the term גרושה [GRUŠE], that is a substantivized (female) participle derived from the verbal root גרש [GRŠ]. This root possesses the basic meaning of ‘to expel (out from)’ - for an example of this meaning, see its use in Isa 57:20. Granted, we cannot find here a specification that indicates us the ‘degree’ of ‘expulsion’ of a married woman. But consider, please, here God is speaking to priestly men, giving them some restrictions about who they may marry. You ask: “Why is it not translated ‘driven out’, ‘sent away’, ‘separated from her husband’?” Can you imagine that a priest may marry a married woman, ‘expelled’ by her husband without a legal base to do so? A man (even more so a priest) that marry a woman in that illegal condition found himself in an adultery condition, and so, liable to be death sentenced. Clearly, the social and lexical context oblige us to specify the ambit the Bible writer did intend in these passages, namely, the marriage Mosaic legal disposition.
We’ve don’t worry about the lack of specification in this passage. In English language also there’s an habit to shorten a more longer (and more correct) phrase. For an example, today English-speaking people say ‘to serve a summons’, that is only a shortened manner (= with less specifications) to say ‘to serve a summons to appear’. Here, the term גרושה [GRUŠE] is clearly related to a legal action of divorce, even it is not followed by an apposition (specification).
So, in this passage we may sense the nuance of ‘to make a clean sweep (of a marital’s relation)’, in other words, ‘to divorce’. Consequently, in its proper context, גרושהindicates ‘(a) divorced (woman)’.
As regards Deu 24:1, we encounter:
כריתת ספר [SPR KRITT], which – excluded the first term [SPR] that means ‘book, scripture’ - is derived from a different verbal root KRT (כרת). The basic meaning of it is “to cut off, strip off, remove”. According John Parkhurst, from this root (stem) was derived the following chain of terms: ‘curtus’, Latin [for ‘shortened’, ‘curtailed’, ‘cut-off’] > ‘court’, French [for ‘short’, ‘brief’] > ‘curt’, ‘curtation’, ‘decurtation’, ‘curtail’, ‘curtlass’ (English [XVII c.] for ‘cutlass’ [see The Short French Dictionary, by Guy Miège, 1690, London]). Here is described a pondered real legal action, not an wife expulsion sprang from an impulse. In fact, the action of her sending away (שׁלח) is subsequent to the action of to put in her hand the ‘scripture of divorce’ (כריתת ספר).
I hope these notes will be useful for your research.