The note on the page clearly states that the calendar is based on the decisions of the High Judiciary Council of Saudi Arabia as opposed to any other country. So there is no notion of right or wrong here - I believe Allah SWT will decide that for all the differences between Muslims.
The API currently supports 4 methods for the Islamic Calendar Calculation - https://1x.ax/islamic-network/libraries/calendar. I'm not aware of any other methods for the calculation. There is also an interesting paper on the subject at https://ulilalbabinstitute.id/index.php/J-CEKI/article/download/7359/6499, but then this doesn't solve calculation issues, unless we maintain some kind of table per country based on sighting (which is rather tedious).
What I can offer currently is updating the calendar on the AlAdhan App to be allow you to choose from the various methods in the API, and if you choose mathematical, to adjust the calendar.
If that will help you, please let me know and we can look into enabling that ASAP.
