David Hare-Scott wrote:

I would not think it is a herb in the botanical sense as it has woody stems
and doesn't die down seasonally, it's a herb in the cullinary/medicinal
sense.

Indeed, this distinction was part of the reason for me asking.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting