This question is mostly about authentication. I would like to have a web page (not on the cloud) that would initiate the sign-in process to the Atlassian Cloud for a specific user. From the Atlassian sign-in web-page it is clear that some form of REST API exists that is used for signing-in, but cannot find any documentation.
I don't believe there are REST APIs for authentication. But you can find the Atlassian Cloud Developer documentation, and specifically how they handle authentication here:
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