I'm not denying these are different plants. I merely said, they were somewhat similar. That's all.They are not the same plant, which was the point I was trying to make before. Bay oil from the Caribbean smells different from bay laurel from the Mediterranean. It smells spicier, with notes similar to allspice or nutmeg.
Bay laurel (on left), aka Greek Laurel (Greeks used it for making wreaths for athletic contests) is a shrub native to the Mediterranean.
Order: Laurales; Family: Lauraceae; Genus Laurus; Species: Laurus nobilis
Bay tree (on right), aka Spice Tree is native to the West Indies.
Order: Myrtales; Family: Myrtaceae; Genus Pimenta; Species: Pimenta racemosa
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They are different plants, as you can see.
You could always send an email to Italian Barber. They might be able to resolve some of those assumptions you have made.