"I am EC'ing my 3-month-old. Currently, I just use cloth diapers - a snappi'd prefold at home, and a prefold in a Bummis cover when outside. It's OK at home, but a little bit of a pain when out and about - taking it all off and putting it all on again. So I want to look into training pants, which I feel like would be easier. My main question - what's the advantage of a partly-but-not-fully waterproof pant like the Shi-Shi-Nix (perhaps worn with a wool or fleece cover) versus a fully waterproof pant like the Bongo Baby pullup...? Also, would something like the Shi-Shi-nix fit under a wool cover like Lanacare/Aristocrats/MamaBless?"
I definitely recommend moving to training pants when you're in the EC'ing groove, as they are infinitely more convenient to use (and less laundry to wash) than diapers. If you're still having poop misses, a side-snapping trainer is a good one to choose, but if you're down to only a few pee misses a pull up trainer is the ultimate in ease and trimness.
The advantage of a partly waterproof training pant such as the Shi Shi Nix, Eco Nix or Little Beetle Learner is that it will absorb a full pee without getting outside clothing wet (provided you change it right away) but it's still relatively easy to tell when you've had a miss because it will feel damp on the outside. This enables you to to keep in tune with your baby and communicate about misses promptly but not have to be wiping up puddles.
A fully waterproof training pant like the Pull Up Pocket Pants is useful when you want more protection for outside clothing in the case of not being able to change your baby right away, such as when you are out and about or during the night if you are not EC'ing at night. The wool or fleece trainer covers that I have designed are a very handy and economical way to add more protection to your stash of non-waterproof trainers, so you don't have to buy (as many) separate waterproof trainers. Also, many people do not like the PUL outer of many waterproof pants and prefer to use the more natural wool cover for the extra protection.
My trainer covers are cut lower in the rise and are trimmer than regular diaper covers. They will fit over many types of trainers, including Shi Shi Nix, Eco Nix, Little Beetle Learners, Papoose, Gerbers, Imse Vimse, Bright Bots and other trainers of similar cut. (Note: I will also soon have my own version of a bikini cut side snapping trainer similar to Poquito pants which will also fit under my covers. And, I am also currently working on designing my own version of a pull up pocket trainer since Bongo Baby is no longer wholesaling these).
You can certainly use a cover such as the LANACare, Aristocrat or MamaBless wrap over a trainer. We found with the LANACare (which I used before I thought to design a wool trainer cover) that we could size down when using over a trainer. It was still fairly long in the rise, but did the job nicely, if being somewhat overkill. It's a trimmer and softer cover than Aristocrat, which is why I carry it. Although the Mamabless wrap is beautiful and soft, I wouldn't recommend it for use over trainers because it is cut very large and is quite stretchy. It's really designed to be used over fairly bulky cloth diapers. Plus the wrap/tie design makes it a bit cumbersome to take on and off quickly for EC. I recommend it more as an overnight cover.
Hope this proves helpful!