It can certainly be confusing but the truth is that none of these factors define a knitting or weaving yarn. These are only conditions of the yarn that should be considered before knitting or weaving a project.
If you are in doubt of whether a yarn will make a good knitting or weaving yarn, first consider how you will use it. For example, a stretchy yarn might not make the best warp. But, it might be used for example. Or, if in doubt, and you want to use the yarn, use it as weft.
If you are using a hand dyed yarn, consider that it has been fulled some and sett it closer than a mill dyed yarn. If knitting, knit a swatch with 3 to 4 sized needles and wash the swatch to determine the needle size to use for the finished product.
In addition to how you will use it, consider the qualities you want in the finished product. Would you like a scarf that is lightweight or heavy, warm or cool accessory, nice drape or very light? Qualities of the finished product are largely determined by the fiber content of the yarn and structure of the fabric.
For a lightweight scarf, go with a laceweight yarn. For something a little heavier, a fingering weight and even heavier, a DK weight. (If you go any heavier, you're going to have a saddle blanket around your neck. Ask me how I know!
For a lightweight, warm scarf, consider a protein (animal) fiber such as wool, silk alpaca, etc., or combinations of the same such as Jaggerspun Zephyr Silk/Merino. Melody Superwash Wool or Harmony Laceweight Merino are also good choices for warm, lightweight scarves.
For a lightweight, cool accessory with nice drape, tencel is a good choice and can be used for both knitting and weaving warp or weft. For something a little lighter, knit or weave with 5/2 perle cotton.
By considering the structure, any of the examples above can be knitted or woven so that you have an airy scarf or one that offers more warmth and drape. Choose knitted or woven lace for an airy scarf, stockinette or twill for a heavier scarf. Change the gauge or sett to suit the fabric you desire.
Knit or weave with the yarn of your choice and enjoy the process!