I think that the stories about Faturan have to do with the lack of knowledge about very dense thermosetting material like phenolic resin. Since the native color of a lot of phenolics is a warm amber color, you can see how it was immediately attractive as an amber substitute. But, it clearly is harder than amber or typical plastics, and machines nicely and wears very well. Heat will change the color as Floor mentions - phenolic will continuously darken as it's heated until it becomes a very dark mahogany / brown. A lot of phenolic resin was dyed, and the dark red colored beads could be dyed as well as heated. Fillers could be added so the final material could be opaque and wood-like as well.

So, I surmise that people who did not understand phenolic resins made up stories about what it was made of, and due to the lovely amber colors, thought that it had to be related to, or made from, real amber.

As far as old vs. new phenolic resin - I have limited experience at this point, but the older beads seem to give a much stronger yellow test result with Simichrome than new beads. There is a degradation product that forms on the surface as the phenolic ages, and there is some reaction with the Simichrome that produces a mustard yellow color. At this point I don't know if the old phenolics had slightly different chemistry as well. This is something that is an ongoing research subject of mine and I'm currently reading old patents and technical papers to gain an understanding of the production of phenolic resins in the inter-war period. And I also need to figure out the darkening of the old phenolic pieces and what chemical substance is reacting with the Simichrome (also works with 409, but gives a weaker color).