Yes, "incense" generally means frankincense and other resins when you're speaking of the fragrance note. I agree that descriptions should be more accurate. But that's what you should be looking/smelling for in various forms(e.g. in Nemo, Passage D'Enfer, Kouros).
What you're referring to as a delivery method is a common way of referring to
the act of *burning* incense sticks. Yes, other scents (lavender, rose, "moonlight", B.S., etc.) are used in the burning form but that's not necessarily incense. Most of the time it's not even scented incense and there's really no incense in them. They're just smoky alternatives to air fresheners. It's only a common use the word to describe all burning sticks. Like using the word "Kleenex" when you want to say "tissue."
True frankincense naturally comes in [dried] drops and is burnt over coal.