You seem to be making two points here:
- That people colloquially call it a tree, so don't "correct" it. We know what they are talking about.
That seems fair enough.
But then you seem to say that:
- There is no definition of what a tree or herb is so you can't separate the two. The term is used to describe non woody growth that dies back (among other things). How can you say the word herbaceous is not a testable definition?
I don't think it is pedantic at all. I have a banana patch, and I certainly consider them bushes instead of trees. I think people are interested in finding out the trivia that a banana tree is not a tree, but an herb.