From Nature Neurology Reviews (bold is my emphasis):
What is the 'official' publication date?
Many journals, and most abstracting and indexing services (including Medline and Institute for Scientific Information) still cite the print date as the publication date. This is an evolving standard, however, and the trend now is for publishers to state both the 'Online Publication Date' and the 'Print Publication Date'. Nature Publishing Group currently publishes both dates for its own papers, and we hope that the scientific communities and abstracting and indexing services will recognize these dates.
For the time being, the reference lists in our papers will continue to follow the standard convention of citing by print publication details if they are available. We expect, however, to review this policy regularly as community standards evolve.
For legal purposes (e.g. establishing intellectual property rights), it is our assumption that online publication will constitute public disclosure. This status is, however, for the courts to decide; our role as a publisher is to provide clear documentation of the publication history, online and in print.
In summary, it seems like the print publication date was originally the standard, but now it is becoming more common to treat the online publication date as a form of official publication as well, as online publication becomes the norm. It's probably up to the journal how they exactly define official publication for purposes of repository deposits, however, and you will have to check with the journal for clarification if it is not already clear.
The US National Institutes of Health requires that NIH-funded research must be archived in their PubMedCentral database 12 months after "official publication." Their FAQ on the matter defines this as follows:
NIH determines the official date of publication for the public access policy based on information received from the publisher and the National Library of Medicine (NLM). The official date of publication is listed in the PubMed citation display for a paper immediately after the journal title abbreviation.
NIH uses the official date of publication for determining the public access compliance status of a paper and calculating when a paper should be made public on PubMed Central. An "epub ahead of print" date for a citation in PubMed is not considered the official date of publication, and these papers are still considered in press.
Note that when only partial publication dates are available (e.g. Month and Year, Season and Year), NLM calculates the date as the first date of that time period (e.g. March 2013=March 1, 2013). See http://www.nlm.nih.gov/bsd/licensee/elements_article_source.html for additional information about NLM dates.
Based on these guidelines, it would suggest that it is the print publication date that is important, but it also seems like they leave it up to the publisher to some extent.