A walking stick can be any or all of the items you mentioned. Though these days with travel restrictions being what they are most people have a pole, stick, or cane that folds into smaller segments.
I have a decorative cane that folds into three sections and fits into a 6" x 12" pouch.
I've seen ski poles, intricately carved canes, victorian walking sticks, even a cane made out of pcp pipe that pulled apart and fit into the gent's carry-on. Airlines want those items to take up as little room as possible as well as store easily.
You can find a variety of poles, sticks, canes at your local sporting goods store, pharmacy, or online.