Twill is a weaving pattern. Cotton used in twill weaving is a staple commodity.
Denim is fabricated from cotton twill yarn.
Cotton cloth us generally translucent, unless the weave is twill or denim, and unless the cotton yarn/thread is thick.
Compact cotton is a high-quality yarn. It is made by processing extra-long staple cotton with a compact spinning system. The process creates a strong yarn with very little pilling.
Twill is a weaving pattern.
Poplin has a fine yarn with thicker yarn across it. Similar weight to Oxford. Twill has diagonal weave.
"Twill", like real velvet, is a type of weave of fiber threads. You could have cotton twill, all right, but you might also have polyester/cotton twill. In theory, you could have twill woven of yak hair.
Yarns that are spun from short fibres, like cotton. The staple is the length of the fibre, so you might say 'Egyptian cotton is better quality than Indian cotton because it has a longer staple.' Many artificial yarns are extruded, that is made like squeezing toothpaste from a tube, so they don't need to be spun to make a long thread.
Tencel is a staple fibre. That is it is cut into staple lengths during production so that it can be processed into yarns and nonwoven fabrics on conventional textile processing machinery. The cut lengths are similar in length to cotton or to wool. Continuous filament Tencel yarns are not produced but in theory could be. In a filament yarn all of the fibres in the yarn have the same length as the piece of yarn.
Cotton twill does shrink a little when first laundered. If you are using cotton twill to sew clothing you should launder it first to allow it to shrink before you begin. The amount of shrinkage you get depends on if you wash the fabric in hot, warm or cold water.
Woven cotton is exactly as it sounds. It is cotton that is woven and as such, has a flat 'finish' to it. Twill is made in such away that there are ridges, similar to corduroy, only usually much smaller. Also, the ridges run diagonal to the selvage, instead of parallel as it is in corduroy. If you were to compare quilting weight cotton, which is woven, with a twill cotton suitable for apparel, the twill usually will seem a bit sturdier. When I make purses from cotton fabric, I use quilting weight for the majority of the bag, but twill for the handles and the bottom of the bag, because I feel it has better strength.
Individual fibers are spun into yarn.
Generally, cotton wool is not spun into yarn per se.Cotton is cleaned then either fabricated into cotton wool or into cotton yarn. Cotton yarn is used in making jerseys.