Let me make this clear -
What is being proposed will have precisely zero jurisdiction over how the individual factions operate. What it will concern itself with is Jedi matters - that is, "Jedi", "Rogue Jedi", and so on - and recognition that different kinds of Jedi are needed for different things. One mindset alone will not be able to address all issues, and that each faction in each sector of the light-leaning galaxy appears to have a defined role, and it would be foolish to demand that they set aside what they already do well. This is how I see things working. There are so many things that each light-leaning group does, and does well, that are invaluable, and crucial to the formation of a full-bodied, operational Order.
1) The SSC, for example, has a higher percentage of healers, pointing to their aid-based role, and younglings, the next generation of those that fight the good fight now. They also maintain a presence around the traditional Sith worlds. This group is protective, nurturing, and watchful.
2) The GA is the front line. This is where the focused fight against the darkness continues. You will find the highest percentage of militaristic Jedi and warriors here, as it should be. In the days of old, when war was present, the Jedi focused heavily on training in lightsaber combat, and more besides. This group is active, aware, and firm.
3) As for the formerly Republic-Exclusive TJO, their minor-faction presence is still too new for me to assess the role they do or could fill. Many of its members are Republic-loyal, and that is their right, for every corner of the galaxy needs a lightsided presence - the Republic moreso, given the sheer numbers they've suffered in loss. Do not begrudge these Jedi for still doing their duty, while this group is finding its identity after being so very much tied to the politik of the Republic, all these IC years.
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What I'm trying to say is, no one interpretation of what a Jedi is, between these groups, is wrong. Rather, they are all right, and not one of them being moreso than any of the others. Between us all, we have the warriors, healers, scholars, and so on, to do the job that needs doing. In essence, we are all capable of being cogs in a well-oiled, diverse machine.
This is, in my mind, what we're going for. All we have to do is want it, work together, and make it happen.
Keep your eyes out for the next Convocation, coming to a thread near you!
EDIT: Thread should be up later today. I have it started, just... I've run out of time, this morning!