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Yes and no.
Your image is downloaded from Photobucket and uploaded to Amazon S3. You get a new url that references that image directly on S3 and that link isn't going anywhere. A lot of topical forum data is captured in these existing images, so it's important to be able to recover them long-term.
Yes. Your images live safely in Amazon S3 and are no longer associated with Photobucket. Your original image still lives in Photobucket though.
I built ImgZeit to provide fast, no-nonsense image hosting for people like you. It's not quite ready yet, but should be available in the next few days.
I'm Nic Aitch, a software engineer in Nashville.