I've started several longer posts, but keep putting them aside in favo(u)r of topics that I can whip up a post for with very little research. I'd claim that this is an effect of having an 11-week-old child, except the real truth is that I'm congenitally lazy. I'm afraid that little Grover has inherited this from me, since she usually can't be bothered to burp (orig. AmE--considered slang in BrE according to the OED, but I think that info is out of date) after eating. But she has the good fortune to be gorgeous, which makes laziness a workable lifestyle, since everyone therefore has infinite patience with her. My laziness just causes people to roll their eyes and wonder aloud how I got so far in life.
But out on a walk today, after mocking an innocent bystander's footwear selection, Better Half noted the beauty of another bystander's walking stick, which led me to abandon linguistic-analytical subtlety and do another simple 'they call it this/we call it that' post.
So, say walking stick to me with my American ears on, and I imagine something like a staff--a big stick, possibly picked up while walking in a forest, used by a hiker (or BrE rambler) who wouldn't normally require that kind of support for day-to-day walking. (See photo here.) It probably wouldn't have a handle. But walking stick is what BrE speakers call what AmE speakers call a cane--a stick, like the one to the right, with a (usually curved) handle and often with a rubber anti-slip bit at the end, used by people with (BrE) dodgy feet/legs/knees/hips/ankles. Very often, walking stick is abbreviated to stick, as in Could you pass me my stick?, which was said by my hospital ward-mate last week. (Yes, if you couldn't guess from the last post, I was in (the) hospital again last week.) I asked Better Half if he'd ever use the word cane. First he came up with (AmE--but making inroads in the UK) candy cane, then he supposed that he might use cane for a walking-stick-as-accessory, for instance, as carried by a male Victorian opera-goer. So, in my AmE dialect, canes are for people who can't/shouldn't walk unassisted and walking sticks are for the able-bodied, whereas in BH's BrE dialect, the stick is for the disabled, and the cane is just for decoration. That said, all the photos on this post are taken from this British company's site, and they do use cane, but only for the type that has four feet--they call it a quad cane.
But sticks/canes are not the only differently-named ambulatory aid. If you're even less steady on your feet, you'll need a walker if you're an AmE speaker, and a Zimmer frame if you speak BrE. The latter is a proprietary name from a London company. The former is not marked as AmE in the OED, but I've only ever heard Zimmer frame used here (and I have heard it a lot, as Better Half's roommate when I met him--his grandmother--used one). Back on Mobility People's site, however, one particular model is called a walker--possibly because it is not made by Zimmer and it would be somewhat nonsensical to talk of a CASA Zimmer frame. Kind of like talking about a Canon Xerox machine--you might say it, but the people selling the Canons had better not.