A quickie: Order?

I am certain that smarter people than I have and will exegete the latest splinter group off of PC(USA) and its intentions.  Myself, I'm rather curious about their chosen name: the Evangelical Covenant Order of Presbyterians.  (ECOP?)

Order?

I'm trying to think of what it means when a group like this calls itself an "order" instead of, say, I dunno, "church" or "convention" or what have you.

So, to rack my brain briefly: what other uses of "order" in such context, as the name of a group (not necessarily religious, I suppose, but certainly can be) come to me off the top of my head?

The Order of the Phoenix -- my first Harry Potter reference in the blog, I think, but maybe not.  Yet, as far as I can see, it's ECOP which is more obsessed with purity in this case, which would be the reverse of how things were in the Potter movies--it was the foes of the Order of the Phoenix who wanted to purge out humans, "mudbloods" and other impure types.

The Order of Kappa Alpha -- no mere fraternity, mind you, not like your mere Sigma Chi Omega or Delta Kappa Epsilon or whatever.  These are also the guys who get all excited about Robert E. Lee and such.  Hmm.

The Order of the Knights Templar -- one of many such orders in medieval Christianity.  A military order, heavily invested in the Crusades.  Also found in modern freemasonry, and was this term also tied up and appropriated in those goofy Dan Brown books, or was that something else?  Hmmmmm.

The Order of Lenin -- that was a medal, which I only remember from a line in The Hunt for Red October.  But I suppose it would have referred to the company of those who had received that honor, yes?

Order of the Eastern Star -- a fraternal organization, vaguely Mason-related as far as I can tell, but I do remember they once gave me a grant for college.

OK, so maybe not that much help.  But I have this stubborn idea that words mean things, and I cannot but insist that this particular choice of word for this group denotes something about it, that perhaps its members themselves may not even realize.