Someone in Tamil Nadu says a festival falls in Aadi. A cousin in Uttar Pradesh says the same festival is in Sawan. They are describing the same day. The difference is which lunar-month system each one grew up with.
Two ways to cut the same moon
A lunar month is one full cycle of the Moon's phases. You can mark its edges at either turning point.
The amanta system, common in the south and west, runs each month from one new moon (amavasya) to the next. The purnimanta system, common across the north, runs each month from one full moon (purnima) to the next.
Both cover the same 29 or 30 days. They just start and end at different points, so their names fall out of step for half of every month. During the bright fortnight (shukla paksha, new moon to full moon) the two systems agree on the month name. During the dark fortnight (krishna paksha, full moon to new moon) the purnimanta name has already moved on to the next month, while the amanta name has not.
So for roughly two weeks out of every four, the same date carries two month names, one about a month apart from the other.
Diwali, in both systems
Diwali's main night is the new moon at the end of a dark fortnight. In the purnimanta reckoning that new moon closes the month of Kartika. In the amanta reckoning the same new moon closes the month of Ashwina, and Kartika begins the next morning. Neither is wrong. A north Indian calendar prints "Kartik Amavasya" and a Maharashtrian one prints "Ashwin Amavasya" for the very same night.
Why the festival date does not move
Festivals key off the tithi, the lunar day, not the month label. Diwali is the new-moon tithi in the Ashwina–Kartika dark fortnight; that tithi happens on one specific civil date, and everyone who uses that rule lands on it together. The month name is a label applied afterward. Change the labelling system and the date underneath stays put.
Where dates genuinely differ between regions, the cause is almost always the sunrise rule — the tithi running at sunrise in Ujjain can be yesterday's tithi in New York — which is a separate story.
What Chakra shows
Every festival page and daily panchang prints the amanta month as the primary name and the purnimanta name beside it whenever the two differ, so you can match whichever calendar your family keeps. The tradition presets go a step further and adjust the festival list itself for the six regional conventions Chakra supports.
Adhika Māsa — the leap month inserted every two to three years to keep the lunar year aligned with the solar one — is detected and flagged in both systems.