liners are basically something to catch the poo, so typically a thin piece of paper (nice paper, not 80gsm A4!) or fleace.
Inserts (I think) are pieces of fabric that add absorbancy, for eg nighttime use. (though that's also called boosters, so I may be confusing myself and everyone else 
.. but it might be that inserts are only used with pocket nappies)
We used Pop-in nappies from Close Parent to great effect from about 1month until age 3 (when potty training finally took). We mostly had 1st generation bamboo ones, with a couple of the 2nd gen microfleace ones. The bamboo ones are much more absorbant, but heavier to carry in the nappy bag. The microfleace ones would leak after about 3 hours of DD in the nappy if we didn't notice the lack of absorbing gels in real nappies means that you don't always smell the urine, and if your child doesn't seem to care whether or not their bottom is wet... I have no experience with the 3rd generation of this system so cannot comment.
We also had a couple of Fuzzi Bunz which I LOVED but which probably wouldn't work for your DH as you need to take the absorbent insert out before washing. (and when yours gets to that blessed potty training time, I highly rate Happy Henies with the poppers on the sides- these are now our night nappies)
When they say "do not tumble dry the outers" they mean "if you tumble dry the outers, they will be RUINED and will NOT catch any pee at all" thanks, Dad
For a bucket system, we used a 35 litre bin and 2 mesh laundry bags from Lidl (no need for expensive "nappy specific" stuff!) This held about a day and a half worth of nappies at first, and 2 days when she was older (probably would have taken more than that at an older age, but we had a good system of laundry: a load a day, alterating between nappies and other things. underpants, etc would sometimes get chucked in with the nappies) Things do wash better if you try and tip the nappies out of the mesh, but that's not super-critical.