Just tell them your situation has changed, you have made certain arrangements and now will be happy to start earlier if they wish to onboard you earlier.

At the old place of employment, you probably had a contract that obligated you to give a 2 weeks' notice. That did not obligate the employer to actually keep you working for those two weeks. The purpose of the 2 weeks' notice requirement is to give the employer a chance to backfill for you or at least arrange for a transition plan. If the employer does not need that time, the employer is well within their right to have you not show up for the entire 2 weeks. Everyone understands you, your new employer understands it, so there is nothing alarming or unusual in telling the new employer that you will be happy to report to work earlier than originally agreed upon.