It's very easy to get confused between mordant and modifier. This isn't helped as many authors, um, may not have the best grasp on the subject. It took me a long time to learn the difference. Finally, a member of my spinning and weaving guild took me to one side and explained it. She's been dyeing and working with fibre arts for almost twice as many years as I've been alive. She's mentored some of the current stars in the fibre arts community. Once I got the basic distinction, it was easier to tell what is a mordant and what is a modifier.
A mordant affixes the dye particle to the fibre. A modifier alters the colour. So far so good... right? You can stop there if you like.
We can often get a colour without a mordant, but this is usually not 'fast'. Basically it comes out in the wash or when exposed to light. Mordants make longer lasting colours. Some plants contain enough mordant on their own: rhubarb and onion skins for example. However, a mordant can brighten the final colour considerably, like this:
Without a mordant, the colour was a muddy green-orange, but the yarn in this photo was with alum as a mordant, which made a lovely warm orange.
Different mordants can alter the colour, also different fibres require different mordants. Cellulose like cotton and linen, need something like tannin. Whereas that does little good with a protein fibre like wool which likes something like alum as a mordant.
So a mordant can act as a modifier and a modifier can act a little bit like a mordant. Iron is a modifier and makes a darker, almost grey colour. Copper makes the colour brighter.
Going from this list:
stale urine - the main active ingredient is ammonia, this is used for cleaning and in lichen dyeing, fermenting, and activating some of the dyes. It is also used in harris tweed lichen dye but a method not fermented.
salt - modifier
vinegar - modifier
wood ash in solution - modifier with some mordant qualities - basically it's a week lye - also used for cleaning
oak galls - a source of tannin, so mordant with cotton, modified with wool
raw alum - mordant for wool, modifier for cotton, however, not sure why it would need to be 'raw'. This is new to me.
water in which rusty iron has been soaked - modifier
willow or oak bark - tannin
copper pieces that have been soaked in ammonia for about 2 weeks. - modifier with some mordant qualities.
I'm over simplifying.
There are other mordants and modifiers, but these seem to be the most common. I really like Rebecca Burgess's book for an introduction into basic natural dyeing. My lichen dye books are somewhere in my massive sorting pile of books right now, but I'll see if I can dig them out.
PS. Lichen can be used as a mordant as well as a dye. For the 'best' results, the method can get complicated involving ammonia and fermentation for months, however, many lichens also mordant by boiling.