You can access a previously processed request using the Harmony job id - it's staged in a Harmony S3 bucket for 30 days.  But yes, if you run the request again from the notebook, you are making a new request and it will create a new Harmony job.
I will try to find instructions for sending Harmony output to a user defined S3 bucket.  I've read that it's an option, but haven't looked for actual instructions yet.
Originally posted by @mikala-nsidc in #399 (comment)