First of all, the basis for the look I ended up with is rooted in headcanon, and headcanons will vary, but even if you don't agree with my interpretation of Shivan technology the effects ought to work fine as simply arbitrary pretty effects.
Now, when trying to come up with visuals for Shivan weaponry, I had a few concerns. One is that the Shivans are supposed to be advanced and powerful. So, one approach would be to give them things that look like supertech doom weapons.
But, problem: the fact of the matter is, in terms of gameplay, Shivan weapons (at least the fighter weapons) are very unimpressive. So I could make the effects smaller and less spectacular to reflect that, but that doesn't seem ideal either.
One other thought I had was this, and this applies to many of the other weapons, too: I want to provide space for mods to escalate into. That is, various mods dealing with the Shivans will no doubt give them more impressive weapons. Those weapons ought to have matching visuals, so I want to be careful to make the visuals for these retail weapons somewhat proportional, in a sense -- so that the mods have somewhere to go with their effects.
Now, in my interpretation of the Shivans, the reason so much of their tech is oddly weak is that the ones we encounter in FS1 and 2 are in something of a degraded state. BP provides a similar excuse -- there, the Shivans are deliberately non-optimized, and so they have weird vestigial technology lying around that isn't implemented properly.
So that gave me my motif: the Shivan weapons should look like supertech doom weapons...which aren't working correctly. So: in that early screenshot, the Shivan PoV one -- the main effect there, with the rings and the flash bits, is not the projectile. It's sort of an extended muzzle-flash. The idea is, the weapon is supposed to fire this ball of concentrated hurty-juice, held together by some sort of gravitic containment field.
But the field generators are broken, or not aligned properly, so the projectile shoots out -- and then the containment field shoots out too slowly to catch up, fails, and breaks apart.
The projectile, meanwhile, goes shooting toward the target -- but it's leaking hurty-juice all over the place, so it's trailing this inconstant gaseous effect. Then it hits the target, or dissipates, and blows up in a puff of gas. Still pretty destructive, but not as destructive as it was supposed to be.
I think this provides a nice balance of visuals that look good while not seeming out-of-place for the rather weak weapons that are the Shivan primaries.