I don't find the "views" of First Aid and Medicine in the BBB and Augmentation to be different in any significant way. Augmentation's mention that "Medicine is also used for treating long-term illnesses, including the effects of diseases and toxins" goes on to point the user towards the section of the BBB governing using the Medicine skill to synthesize antidotes to a toxin. While he can't synthesize an antidote, a First Aid practitioner can still administer one if he knows what toxin he's treating... determining what someone's been poisoned with is within the realm of First Aid per both the BBB and Augmentation.
The game is actually pretty clear about defining what a "set of wounds" is, in response to your post of a couple of hours ago. A "set of wounds" is any and all of the damage you've taken since the last time you recieved medical attention.
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The crux if the issue to me is this,, The FAQ sets Drain apart from all other damages/wounds/trauma.
The question asked in the FAQ was "Can Physical Drain be healed by magic?" The answer was "No." They go on to explain that "Damage from Drain must be healed by regular mundane medical care and/or rest." We've only got three kinds of healing in this system... magical, mundane, and resting. How a statement forbidding magical healing of drain damage can get torqued around into "First Aid can't heal drain damage." is beyond me.