The tanning process is quite complex but you can find hybrids of the ones you mention above. The pictures of my remnants are a mixture of all 3.
The oil gives it a distressed look, Chromium is sometimes required in the process to import hides and it allows the dying process to occur much faster.
In certain processes oil and vegetable tanning can be interchangeable, or the same although in this case it probably refers to the distressed finish I mentioned above.
I would disagree you can’t laser chrome tanned hides as this is what most large manufacturers use but the fumes need to be controlled.
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