Industry, for the most part, has one set of definitions for normalizing while there can be as many definitions among bladesmiths as there are bladesmiths. The idea behind normalizing, as has been stated is to evenly redistribute carbon and structures and resulting stresses in the steel. Grain refinement is often more a reference to homogenous grain size, whatever that size may be, and for good reason since uniformity can often be more important than actual size. True normalizing requires heats high enough to put the material in full solution and thus is often in the 1600F 1700F range. Air cooling is important for actual normalizing in order to keep things into solution evenly on cooling, and to be certain things cool uniformly. So to actually normalize blades the first heat should be well above critical, but as mentioned never yellow or white hot.
Bladesmiths then follow this heat which equalized things with a heats to refine that condition further. These heats are almost universally referred to as normalizing by bladesmiths but should actually be called thermal cycles to be more accurate. The next heat will be to make smaller grains from the evenly sized ones and is done just to critical and then air cooled. Some will quench at this point instead, I often do, to increase the grain refinement, a quench will replace around two air coolings with careful heating.
For any of these refinements, you really only need to air cool to Ar1 (when the magnet sticks again) before reheating, but if you quench you will need to go to room temperature in order to see gains, otherwise you will only be reheating the same internal condition (austenite) again.
Often smiths finish up on a dull read final heat. This can create subgrains, like grain seeds that will sprout into new, finer grains on the next heating. But if one is careful not to allow the blade to go nonmagnetic this dull red heating can also substitute for an anneal, and is preferable for steel with more than .8% carbon.
Annealing is for the expressed purpose of softening the steel and can be a full (lamellar) anneal, where you heat to critical and then slow cool (wood ash, vermiculite oven etc
). Or you can got with a sub-critical anneal which takes us back to keeping the heat below nonmagnetic and softening the steel that way, this is called spheroidizing. Spheroidizing is highly recommended for anything over .8% carbon and also helps keep everything you did in the normalizing, while full annealing tends to undo some of it.