First let me say that you have me at an obvious disadvantage. You have introduced "process thought theology" to me and I have only read what you have posted and one of your links.(
https://clayton.ctr4process.org/files/papers/GodBeyondOrthodoxy-r3.pdf)
Tbh I don't really see "the process" in it though this may due to my admitted ignorance. It seems to me that the whole thing is simply defining what is unknowable and then labeling that god. I would prefer the label be left as unknowable because that is what it is.
I can see two groups to which I think this probably has the greatest appeal.
1)Those that would like to rescue God and so their faith from whatever knowledge they have of the universe that has introduced doubt.
2) Those that have a strong desire to preserve the idea of some degree of freewill.
I will admit that I fall in the second camp. Freewill does seem to be less and less free and I am somewhat uncomfortable living in an increasingly fatalistic universe, but i am not ready to hang my hat on God, magic, or any other self serving concept in order to comfort myself.
Well, first I'd challenge you on the two psychological motivations you pose. Process theology may well appeal to people like that, but it might well be some other motivation -- like a desire to find meaningfulness in life, or even just a desire to understand the universe better. I think I may have been drawn to process thought partly for the former reason, but now I'm in it more for the latter one. A lot of the metaphysics just makes more sense to me than anything else I've heard.
To use the Wikipedia definition, metaphysics is the branch of philosophy concerned with explaining the fundamental nature of being. It's often scoffed at nowadays because metaphysical theories aren't empirically testable; you're reflecting this attitude when you say that "It seems to me that the whole thing is simply defining what is unknowable and then labeling that god. I would prefer the label be left as unknowable because that is what it is."
But the thing is, everyone has attitudes and assumptions about "the unknowable," even if they remain unstated. To quote Whitehead, "Every scientific man in order to preserve his reputation has to say he dislikes metaphysics. What he means is he dislikes having his metaphysics criticized." To go back to my favorite example of a metaphysical belief, I really think that there is an "interior dimension" to all matter, that there is no such thing as fully "dead" matter (although "matter" itself is a problematic concept, which I'll get to later, but you understand what I mean). This is not a testable theory, because it's not something we can access -- even for humans, we can only see the behavior resulting from their mental lives, but we can't jump in their heads and experience what they're experiencing in the way they're experiencing it.
So, what's the significance of holding a belief in "
panexperientialism," to use David Ray Griffin's term? Nothing too specific, but more of a general attitude and outlook toward the universe. As you say, it certainly undercuts determinism. It also tends to steer one a little further away from a purely human-centered worldview -- doubly so if you buy into the constitutive interrelatedness that process thought stresses, that we are a sum of our relations to other entities, not independent of them.
The point is that I don't think people come to process thought purely out of some desperate psychological need/desire. Frankly, we could look at any metaphysical/philosophical/religious position and come up with psychological reasons why the person holds it. But while it's helpful to examine that aspect of things now and again, I'd rather look past motivations and see if the systems and beliefs themselves make sense.
While I'm here, I should also explain the "process" aspect a bit, since you mentioned it, although I'll be getting kinda technical here. Among other names, process thought is sometimes also called "
process-relational thought/philosophy/theology." I've already explained the "relational" part a little bit. The "process" aspect points to a belief in the primacy of
becoming over
being. Western philosophy has always considered
being to be the primary ontological mode/category: there are beings/things that persist and change (this is tied up with the previously mentioned subject-predicate ontology: there are
things with
qualities that can be
predicated of them).
But in process thought,
being is an abstraction, and
becoming is primary. A given thing -- like a person -- is conceived of as a series of momentary events strung together. Each event in this series "inherits" its past event, including its motivations and desires, and influences the following event. Whitehead has a nice phrase: "perpetually perishing." Each event dies almost as soon as it arises. But it is these momentary events that are the "really real things," while an enduring person, for instance, is an abstraction achieved by conflating these events.
This may sound like largely a semantic difference, but it really changes the metaphysical/philosophical outlook and gets around a lot of philosophical problems. A subjective-predicate ontology always had the problem of explaining, for instance, what it could mean for there to be a thing
without qualities. It opens the door to the idea that "substance" and "matter" are crappy categories for understanding the nature of reality. And when you accept that "matter" as a concept sucks, you start looking at the universe in a very different way. And although I am certainly no physicist, my understanding is that modern physics has already done away with matter as a key concept.
Returning to God briefly, I'll say again that for me, God isn't the most important part of all this. Everything I've just explained can be held without a belief in any kind of God. Further, as you pointed out, a particular formulation of "God" might diverge so greatly from what the word usually signifies that it can be seen as an entirely arbitrary naming/identification. I'm very sympathetic to that perspective. If you don't like the word or the baggage that comes with it, then just don't use it. Just talk about the nature of the universe instead.
I'll leave it there for now, as I've blathered quite enough and have probably wandered a little far afield.