I think alchemy is mainly about chemistry, while I agree herbology has nothing to do with magic. I believe alchemy does, because like you say Thom, alchemists tried to make philosophers stones, life elixirs, turn lead - Pb into gold - Au and so on. But they never managed it. With time alchemy turned in chemistry and there is rarely someone who believe these failed experiments and attempts are possible at all. But different alchemy experiments may as well have nothing to do with magic
In fantasy worlds like PS, where magic is present, it is also part of alchemy. Like in healing potions. There is no purely chemical substance, which would regenerate strenghts so fast, like the healing potions do. But i would like to note there, that alchemy is still very different than magic purely from the glyphs. For example I believe the strenghts regeneration process after drinking a potions, shouldn't be as fast as effect of casted spell. There are number of differencies, which should bo present, to clearly distinct alchemy from the ways of magic.
Bilbous is asking a good question about the source of magic in alchemy products. It has originated from use of glyphs in some way for sure. But claiming that with use of alchemy you can extract a glyph from nature to your body is fragile theory. Like it was said, to extract a glyph, you need a carrier. How to say it... a skin may not be a carrier. It is not durable enough. It gets damaged too easly. And you should know what happens if a glyphs gets destroyed.
What you could do, to make your beloved tatoo, is to put the carrier with glyph undewr your skin, like it was suggested. Then make some nice drawing around and done. But i don't know if you would enjoy it^^ especially possible death from infection.