This is a story because people are stupid. And stories like this make people even stupider.
No, the second-born twin is not "older." Any vital-statistics mechanism that makes that fiction appear to be truth had to have been designed by someone who watched nothing but local-television news.
The solution is so very simple: List each time of birth with either EDT or EST. Call it good. Or if the hospital paper-pushers wanted to get fancy, also list the time of birth for each twin with Coordinated Universal Time in parentheses. In UTC (formerly known as Greenwich Mean Time), there is no daylight-saving time confusion.
Now call me back when a mother gives birth to Twin No. 1 on a ship or plane just west of the International Date Line and Twin No. 2 just east of the date line. They'd be born at nearly the same time on different days*, with Twin No. 2 born a day earlier.
(NOTE: Fix is the same, but the circumstance is more interesting.)
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* Depending upon the peculiarities of the local time zones you were in or over.