What counts as a "prior version" in Springer's self-archiving policy?
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.