This pics show the insides of the different ones.
https://www.mother-ease.com/cms/editor1/images/OSD_Fabrics_Staydry.jpg
https://www.mother-ease.com/cms/editor1/images/OSD_Fabrics_Organic.jpg
https://www.mother-ease.com/cms/editor1/images/OSD_Fabrics_Bamboo.jpg
https://www.mother-ease.com/cms/editor1/images/OSD_Fabrics_StretchyCotton1.jpg
https://www.mother-ease.com/cms/editor1/images/OSD_Fabrics_StretchyCotton2.jpg
The top is the stay dry - you can see on the inside there's a synthetic fabric that has a sort of diamond pattern in it I think. From what I remember of the Motherease stay dry fabric, it's almost like a very fine mesh over the top of something else... kind of like the mesh type top layer of a sanitary pad! The cotton ones will feel more like a towel inside and so will the bamboo probably if they're preloved. If they're new, the cotton ones will feel like a normal towel and the bamboo will feel silkier.
What Motherease call the 'snap in liner' is actually a booster (liners aren't usually absorbent; boosters are used to add extra absorbency to a nappy, that why I'd call it a booster!). For an 11 month old I would say you'll definitely need to snap the booster in all of the time, for days. The 'snap in liners' are available in the same fabric options as the nappy, so you might find you have cotton nappies but stay dry liners or something, they won't necessarily match

If the liner is topped with stay dry fabric (the synthetic mesh-like stuff), you won't need anything else for LO to feel dry and comfortable. However, if it isn't stay dry, LO will feel the wet. Some babies don't mind that but if he's been in sposies I think he will need a stay dry nappy. If your snap in boosters aren't stay dry, you can just add a simple piece of fleece - any old fleece blanket cut up will work (do it to around the same size as the snap in liner) and just lay it in, or you can buy them online. The other option is disposable liners but I wouldn't bother unless you feel you need them to help poo clean-up. They don't keep LO feeling particularly dry and mean something you have to throw away. With an 11 month old, you can probably just tip poo off the nappy into the toilet, or wipe it in with some loo roll (no worse than wiping LO's bum!).
You may want to buy some extra boosters but I wouldn't at first, just try what you have for now. An old face cloth/teatowel would work if you need something straight away.
Sorry for waffling. What I think you might need/want:
Wraps and bucket, as you already know
Fleece liners if the ones you have aren't stay dry
Perhaps boosters later on, if the nappies don't last very long