Akshaya Tritiya falls on the third day (Tritiya) of the bright half of Vaishakha. The word akshaya means never diminishing, and the day is traditionally seen as a good one to begin something new, to give in charity and to remember ancestors. Many families do a simple puja, offer water and grain, and some buy a small token such as a coin or a book.
The date is fixed by the Purvahna, the forenoon, which is the first part of the day after sunrise. The festival is kept on the civil day on which Tritiya is in force during the forenoon. When Tritiya is in force in the forenoon of two days, the first day is taken. The puja window in the table runs from local sunrise to the end of the forenoon (or the end of Tritiya, if that comes first).
Why do cities differ? Tritiya begins and ends at one instant worldwide, but the forenoon is a different clock time in each city. A tithi that starts shortly after the morning in one city can already be running at the morning of another city to the east. That can move the festival by a day, and it always moves the clock time of the puja.
A calm note: the day is about generosity and fresh starts, not about spending. If you plan a purchase or a financial commitment, make that decision on its own merits and, if it matters, with proper advice; the day does not make a decision better or worse. If your family or temple keeps a different day, follow them.