You should probably not worry about it.
On the one hand, having solid references really protects you. You made an honest mistake on your resume, and HR at the prospective company will almost always understand that. If they ask, explain to them that you did not realize that the period before the official start date did not count, despite you working closely with the team during that time period. Your references can verify that you worked closely with them, and by taking ownership of it as a mistake, they are much more likely to tell you it doesn't matter, and just ask you to update the copy of your resume they have on file.
On the other hand, not telling them can lead to a situation where they see you failing a background check for lying on a resume. While not dishonest on your part, the official date is what will show during a BG check, and that's what it will compare against your resume date. Regardless of how it came about, false information on a resume is always a terminable offence (though they won't unless they're looking for a reason to get rid of you already, or it involves something illegal/much more major in scope) even if it was not malicious.
I think in either case, pleading ignorance on your part is honest and the best course of action. I would personally inform the HR department of the mix up, and attempt to rectify it immediately. If they found out ahead of time and asked me about it, I would state what you did in your question, and plead ignorance as to the unofficial start date being unacceptable.
I don't think they will put as much weight on it as you have though.