Hi Louise, good that you are thinking about this because good picking helps increase harvests.
You can do it either way, I prefer the one you describe and leaving some stem with its small buds is good in cold weather when it takes longer for new stems to emerge.
However you pick, the stems become thinner with time and then a little woody. Last year I had longest harvests from a "Late Purple Sprouting" variety, because of the wet and cool spring, so a lot depends on weather as well as how you pick it.