When carrying out the float test you need to make sure there aren't any contaminants, the most common being latex adhesive. Before immersing in water, soak the fibre first and expel all air as this could make any fibre float. If the fibre does float, this suggests that it's polyprop or a high percentage polyprop blend.
Burn tests have various indicators to suggest the fibre type. The nature of the ash/bead, the flame, the extinguishing properties and the odour being the principal indicators.
For the way I clean, Woolsafe and Microsplitters, it is not usually important to know which exact fibre I work with, only if it's wool or synthetic. In the real world, with an experienced eye, it would be reasonable to presume that, for example, an expensive looking Axminster weave is going to be a wool/wool blend. I have to admit though, that experience and knowledge isn't infallible. Several times a year I reckon to be caught out by tufted twist piles which I presume to be wool but are actually polyprop in good condition. But as mentioned earlier, because of the way I work it isn't a problem. I merely change from my soft Host brushes to a harder grade. Oh, and I wouldn't apply protector.
safe and happy cleaning:)
Ken