If my paper is accepted in a Springer journal, can I submit my version of the manuscript to arXiv immediately after acceptance? I have read the Springer copyrights and self-archiving policy, but they are a bit confusing. For example in the first of them, you may find this:


  Author may self-archive an author-created version of his/her Contribution on his/her own website and/or the repository of Author’s department or faculty.


And here is the quote from the second source:


  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. He/ she may not use the publisher's version (the final article), which is posted on SpringerLink and other Springer websites, for the purpose of self-archiving or deposit. Furthermore, the author may only post his/her version provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website.  The link must be provided by inserting the DOI number of the article in the following sentence: “The final publication is available at Springer via http://dx.doi.org/[insert DOI]”.


Do I understand correctly that I can put my own version of the manuscript at my own (or my institute's) website immediately after acceptance, but I can submit it to arXiv only 12 months later after official publication? If so, it seems to me it doesn't make sense. If the manuscript is made publicly available just after acceptance at my own website, why cannot it be made publicly available at the same time at arXiv?
The short answer is: yes, staggered posting rights like this are very common - many journals distinguish between your own website, your institutional repository, and a broader repository (eg arXiv, pubmed), with different rules about what version of the article can be posted and when; some have also begun to provide a special category for sites like ResearchGate. 

The underlying answer -

Why on earth do they do this?

On the one hand, this doesn't make sense - the PDF is the PDF and once it's picked up by a search engine, it's going to be readable wherever it's hosted. But consider it in terms of discoverability and scale:


one hundred papers from the journal on one hundred obscure personal sites;
one hundred papers from the journal on twenty well-organised institutional sites; or
one hundred papers from the journal on one disciplinary repository


From a publisher's perspective, the first option is not something to worry about, while the second &amp; particularly third options look quite scary. Remember, what they really don't want is everyone saying "great, we can get this all from arXiv, cancel next year's subscription please". So an embargo period gets attached to the repository copies - many journals (most prominently PNAS) manage fine on a subscription basis while still making older papers freely accessible after a year or two, and so it's well-understood that allowing delayed access in one form or another will not ruin the subscription income.

Now, mass cancellation because of self-deposited papers being available instantly is a bit of a bogeyman. No-one's really shown it would happen on a large scale (and indeed arXiv suggests otherwise); library budgets are not (yet) squeezed enough that we've had to start thinking seriously about it; and in any case "big deal" subscription models often make it impractical to cancel specific titles. But it's looming as the threat and most publishers simply don't want to risk it... so they produce very conservative guidance on what you're allowed to do, and work from there.

Over the past few years, many of these publisher limits have loosened slightly as they discover the sky didn't fall, which I suppose is something. At the time of writing, at a very loose generalisation, policies for non-medical scientific journals are mostly converging along the lines of "accepted MS immediately on your own site, accepted MS in a repository after a year, publisher/proof PDF never", but there are a thousand variations.