A few ideas that pertain (at least) to AE...
I would say a window shade or a large outdoor roller blind is wound or reeled onto a central wooden or metal shaft, shank or roller.
The user rolls or reels or cranks the shade up or down either with a chain or a crank... or by pulling it up or down by hand.
A small device for holding and dispensing thread, particularly in the under part of a sewing machine, is a bobbin.
Bobbin has other technical meanings, but isn't used commonly for anything other than this sewing machine device.
Sewing thread is sold as spools of thread. One uses a spool of thread on top of a sewing machine or for sewing by hand.
A spindle is another word usually reserved for thread and is usually part of some kind of a machine.
Movie film, as well as still camera film, is kept on reels which are put into canisters for storage.
Yarn for knitting is sold in skeins. A skein does not have a core at its center.
General purpose string or twine for wrapping parcels or for kite flying is sold in spools. The core that it is wound on is often cardboard or plastic and may be either cylindrical or conical.
Fabric, as wound for sale, is sold as bolts of fabric. The central core of a bolt of fabric is usually heavy cardboard which may be flat and wide or cylindrical.
The word coil is seems to be reserved for metal wire or perhaps stiff plastic –something that is resistant to having its shape changed– which can be wound or coiled onto a spool. If the spool is quite large, perhaps large than a hand, it might be called a reel. Something that is coiled and can be uncoiled is usually under some kind of tension and tends to retain its coiled shape as it is uncoiled from its spool or reel. Snakes can coil themselves and are then coiled. Garden hoses can be coiled up for storage. A garden hose can also be wound or coiled onto a standing reel, a device for storage which might have a hand crank.