If you really want to do it yourself then you will need to stop using simple on-line dictionaries.
There is a very good reason that access to OED on-line costs quite a lot of money ($300/year or thereabouts), it's because it is jam-packed with the history of words.
If you use OED on-line it's easy to trace inure and enure back to their roots, which is the noun ure. I can't find a free on-line dictionary that provides a definition for ure except when it is used as a suffix -ure.
In OED1 ure covers nearly an entire page, inure covers a third of a page and enure about a sixth. Compare that with on-line dictionaries, where you have a one line definition and no history.
I've copied the relevant portions of OED1 for ure, enure and inure here but can't provide an answer as to how enure got stuck as the 'Law' term and inure continued in common use.
f. en-1, in-2 + ure, work, operation, exercise, use, a. F. œuvre work.
1.1 trans. To bring (a person, etc.) by use, habit, or continual exercise to a certain condition or state of mind, to the endurance of a certain condition, to the following of a certain kind of life, etc.; to accustom, habituate. a.1.a Const. to (†unto), inf.
4.4 intr. Chiefly Law. To come into operation; to operate; to be operative; to take or have effect. Often in form enure: see enure v. 3.
en-1 + ure, a. OF. oeuvre operation; cf. poure, poor, a. OF. povre.
2.2 Of persons: To bring by use, habit, or continual exercise to a certain condition or state of mind, to the endurance of a certain condition, to the following of a certain kind of life, etc. Const. to with n. or inf. Now only in form inure, q.v.
3.3 intr. Chiefly Law. To come into operation; to take place, have effect; to be available; to be applied (to the use or benefit of a person). Const. to or simply.
a. AF. *eure, = OF. uevre, euvre, evre (13th cent.; F. œuvre):—L. opera OPERA sb.
I. in ure: 1. a. In or into use, practice, or performance. Often with vbs., as bring, come, have, and esp. put (freq. c 1510–c 1630). Also rarely with into.
c. With reference to statutes, etc.: In or into effect, force, or operation. Chiefly with vbs., esp. put.
III. 4. Custom or habit on the part of persons; wont to do something. rare.
As you can see, ure, inure, enure all have meanings of 'become accustomed to something', like your first Oxford Dictionary link and a meaning of 'to take place, have effect' in a legal sense like your second link.
The only way to track then back to where the meaning of enure and inure split up or became favoured in the legal profession over another meaning is to look at the references given by the OED stretching back to the 15th century and to do that you need to get your hands on either access to OED on-line or use the links to the copies of OED1 provided at this meta post by the user MετάEd .
I've also found that 'The Free Dictionary by Farlex' is also quite good; having access to a number of standard dictionaries, a financial dictionary and a legal dictionary. It also mentions the root of inure as
Middle English, back-formation from enured, customary, from in ure : in, in; see in1 + ure, use (from Old French euvre, uevre, work, from Latin opera, activity associated with work; see op- in Indo-European roots).
inure. (n.d.) The American Heritage® Dictionary of the English Language, Fourth Edition. (2003). Retrieved September 26 2014 from http://www.thefreedictionary.com/inure