Loob, I'd supposed that your verb bath was our bathe; I hadn't realized you had both. Under what circumstances would you bathe your dogs? (Or is it never transitive?)This evening, I had to bath both my dogs . . .
When you wash yourself you can say that you bath (British English) or bathe (North American English) , but it is much more common to say have a bath (British English) or take a bath (North American English).
You can also bath (British English) or bathe (North American English) another person, for example a baby.
You bathe a part of your body, especially to clean a wound.
When you go swimming it is old-fashioned to say that you bathe, and you cannot say that you bath or take a bath. It is more common to swim, go for a swim, have a swim or go swimming:Let’s go for a quick swim in the pool. ◇ She goes swimming every morning before breakfast. What you wear for this activity is usually called a swimming costume in British English and a bathing suit in North American English.
Thank you, Loob! The difference between your verbs "bath" and "bathe" is now totally clear.
The note from the Oxford learner's dictionary is also quite helpful, although one item's outdated: I can't recall when I last heard "bathing suit"; the item is a swimsuit in AE.