I looked at dozens of newspapers, and I am still waiting for other Boonville Advertiser articles, but here are the reasons why I assert it is Joseph Franklin Rutherford:
1) Small town, 4400 people.
2) He always appears in articles using the same initials, but it's not questioned whether it's him because there are other factors such as a) mentioned in relation to lawyers or b) mentioned in relation to the democratic party.
But the same skepticism could be applied here: ONLY THE INITIALS APPEAR. Could there have been TWO J.F. Rutherfords who were involved in laws in Boonville, Missouri, except one was John F. and the other Joseph F.? It's a little ridiculous. J.F. Rutherford, in Boonville, Missouri, involved in laws or the democratic party, is automatically assumed to be Joseph Franklin Rutherford.
However, I found an "Hon. J.F. Rutherford from Pine Bluff" article, and an "Hon. J.F. Rutherford from Boonvillle" article. According to your logic, it could have happened. But no.
You mention that John F. moved from Missouri to Arkansas. Proof of that?
3) In other articles, he is not mentioned in relation to laws or the democratic party, but he is mentioned, always, as "J.F. Rutherford, from Boonville". In many cases, he is noted to be a brilliant orator.
Therefore, since we have:
A J.F. Rutherford in the Boonville Advertiser being involved in a fraternal organization which could help in getting good contacts for a lawyer who wants to run for office, who is involved in public speaking for this organization, it can be safely assumed it's him. Same newspaper, same town, same first, middle intials and last name, similar public activities...
I have not contested your Wikipedia change, since for now I only have indirect proof. But I think that your John F. from Pine Bluff has no bearing on this.