What counts as a "prior version" in Springer's self-archiving policy? Springer's self-archiving policy ( https://www.springer.com/gp/open-access/authors-rights/self-archiving-policy/2124 ) says:


  By signing the Copyright Transfer Statement you still retain
  substantial rights, such as self-archiving: "Authors may self-archive
  the author’s accepted manuscript of their articles on their own
  websites. Authors may also deposit this version of the article in
  any repository, provided it is only made publicly available 12 months
  after official publication or later. (...)
  
  Prior versions of the article published on non-commercial pre-print servers like arXiv.org can remain on these servers and/or can be
  updated with the author’s accepted version.


So if I'm understanding correctly, if you have not uploaded the paper to arXiv or other repositories beforehand, then a 12-month embargo applies before you are allowed to publish the accepted manuscript in any repository. However, if you have uploaded the paper to arXiv or some other repository, then you are allowed to post the accepted manuscript there with no embargo as an update of the previous version.

Not that I understand why such a policy can make sense, but my real question is: how should "prior" be interpreted there? Prior to paper acceptance or prior to the first submission?

That is, for example: is it OK to submit a paper to one of these journals, be asked for major revisions, then do the revisions and submit them to arXiv in addition to the journal, and finally update the arXiv entry when the paper is accepted?
I would interpret "prior version" here to mean any version prior to acceptance, which would include ones submitted for revisions.  Thus, I believe that you are correct in your conjecture: you can submit to arXiv and keep updating that at any time, as long as your first submission is before acceptance.

So, why does such a policy make sense? My take is this: if the manuscript is already out there on arXiv, then it's in the journal's interest to make sure it's the most accurate version.  From a business perspective, for-profit journals probably do not like having things appear on arXiv, but in those communities where preprint circulation and self-archiving are the rule, they don't really have the power to prevent it.