How to find out if a paper is diamond/gold/green access?
if the paper is listed in a journal but a pre-print is available on the authors' webpage/repository then it is Green access. Is that correct?

Not exactly. The paper may be published under a green OA agreement even if the author did not publish it on their webpage, or the author may have shared it illegally. For a more authoritative answer, you should instead look up the journal on Sherpa/Romeo and see if it allows publishing pre/postprints.

Next how do I find out whether a paper appearing on the journal's website is gold access or diamond access? (Basically did the authors pay to make their paper open)

The journal's webpage will usually tell you if there are publication charges, and how much they are. Even if they advertise charges, these may be waived under certain conditions (for instance: agreement with the author's institution; author comes from a developing country). Ultimately, there is no way to pay if two third parties reached an agreement and waived that fee; that is a private contract between them and none of your business. :)
(Anyway, why do you care exactly? Is this to know if you will be charged to publish your future paper in the same venue? Or for a research/statistic?)