I've done a bunch of (what seems like) introspecting a bunch of (what seems like) ruminating. To me, there is a distinct difference between the two, but ironically I think telling the difference requires you to have some skill at introspection.
I could point to concrete differences, such as:
But ultimately it's more like "Rumination has a different flavor than introspection." Sometimes introspection is goal directed, sometimes it's not. Sometimes introspection is repetitive. Sometimes it's not. The rules I can come up with for "what is introspection?" don't feel ironclad.
(Except perhaps that it feels more like listening than talking/acting?)
But nonetheless, when I'm ruminating, my whole body usually feels like it has a particular stance – more stressed, more concerned with action (sometimes rehearsing actions I plan to take later, sometimes just worrying about the state of the world as it is now). Whereas introspection feels more tranquil, like I'm sitting by a lake and observing it.
And maybe (if I'm having a bad day) the lake is full of garbage and the sky is stormy. But my stance towards the lake isn't trying to fight anything, it's just sitting and noting "ah, I see the lake is full of garbage today." Whereas if I were ruminating I'd be thinking "gah the lake is full of garbage the lake is full of garbage I want it not to be full of garbage the lake is full of garbage."
(And yes, in that example the rumination thoughts are repetitive, but I think the key ingredient is beginning with "gah" instead of "ah")
So one of the distinctions folks make within meditation (a special kind of introspection, we might say) to help people focus on the right things for insight is to distinguish between content and structure/process. That is, you're often given the instruction to just be with whatever comes up, but this doesn't mean to spend your time being trapped in it; instead the idea is to see something about what's going on that's elucidated by what came up.
To make this more concrete, say you sit down to meditate and you start thinking about how you felt awkward about something that you said to your friend earlier. Maybe that's not the object of concentration, so maybe you should just drop it, but if you stay with it that could be fine too, especially if it's intrusive and no matter how hard you try to focus on your breath or whatever you keep drifting back to thoughts of the awkward conversation. Whether or not what you are doing is skillful meditation will depend on if spend your time focused on the content of thought (e.g. "oh no why did I say that, I bet they hate me now, how was I so dumb, I always do this, what if I did something different instead, if only I had a different childhood I would be different, ...") or whether you spend your time noticing what's going on (e.g. "okay, I feel awkward and keep thinking about it, what does it feel like to feel awkward? where do i feel it in my body? okay, now why do I relate to it the way I do? what comes up when I ask "why is this awkward"? am I sure this is true? ..."). This seems to be something like the rumination/introspection distinction you mention.
Now of course today's structure/process is tomorrow's content, so this distinction is a relative one to how you relate to your own thoughts. Further, introspection/meditation is a place where it's important to be epistemically humble because our brains often don't know themselves very well and may feed us confused misinformation that we will later see as confused but right now can't separate out from reality, so the conclusions we can draw from introspectively accumulated evidence are necessarily weaker due to the lower confidence we should have in the capta. Thus it makes sense to be cautious about our claims based on introspection, even as they are often extremely helpful for some purposes, like understanding ourselves better and becoming less confused about our relationship to the world.
Your definition of ruminating includes that you introspect on causes and consequences as opposed to solutions. The techniques you mention may include focusing on causes and consequences, but they are very solution-oriented.
If there is a difference in their successfulness, I think that solution-orientedness is why. People who ruminate are thinking about a problem without trying to solve it. That's, frankly, a depressing thing to do. Feeling like you have a problem that can't be solved is almost the definition of frustration, and just reminding yourself of a problem without any sense of moving forward or making progress will reinforce negative thought patterns without accomplishing anything.
By contrast, people who engage in focusing, IFS, and related techniques have a goal in mind. They're not just reviewing the problem and its causes; they're trying to get somewhere. There's an underlying optimism that is being fostered, especially if it works well enough for people to want to keep trying it.
The thing that makes rumination bad is not its topic, but its repetitiveness; the first time one has a rumination-flavored thought it is potentially useful, and is not called rumination.
Consider someone learning a piano piece, and having difficulty with some passage.
If they merely try to play it over and over again, resulting in the same mistakes every time and the same frustration, they fail to make progress.
What they need to do in such a case is think about how they are trying to play it, work out why they are going wrong, and experiment with different fingerings and arm postures. The repetitive practice is still necessary, but it must be directed by an awareness of the possibilities that are available to explore, rather than a narrow focus on something that is not working.
As Edward de Bono puts it, "You cannot dig a hole in a different place by digging the same hole deeper."
For me, what separates mindfulness from rumination is that in mindfulness you observe things and accept them, whereas in rumination you're trying to fight or hold onto something.
Constantly reminiscing a slight is a good way to make it loom large. It's an unwillingness to either resolve the matter and letting it be.
Similarly, fighting some negative emotions (pain, loss, anger) makes them worse when they inevitably breaks through.