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The MSD only sets the year.
So in this case, MSD 2013 (from your first Amex in 2013), if you were to apply in 01/14 your open date on your credit report will be 01/13, free year of AAoA.
That said, you can get the backdate to 2013 *anytime*. I don't honestly think that a single inquiry / new tradeline in January would hurt all that much by July, it may even be a positive with 6 months of payment history on it vs. one additional inquiry (since you won't be at zero otherwise by July anyway) depending what the rest of your credit report looks like; however, a year of backdating isn't worth coming out of the garden for ever. That will always be there, your shot at the best possible auto loan rate won't be.
Always prioritize big ticket items, mortgage >> auto >> credit cards.
>> = much greater than in this case when it comes to relative importance.