Question about goth and gothic, I want to make sure if i understand it right

As I see it

Gothic- is more about the old art and darker climate

Like gothic architecture and books/poems and dark vibes and like interests about it

Goth- is more about the music, appearance, and whole esthetic to express yourself

And both can work together but don't have to, goth people can have also interests in gothic as they're close but doesn't need to

Is it fine?