How is progress towards the Moments of Triumph trackers calculated?

Solution 1:

The accomplishments shown are for the character that has the most of them.

All region chests (except for Mercury and Mars) are for your entire account: if you open it on one character, it's considered open for all characters. This is why getting the chests on your alt updated the count immediately.

For the Adventures in particular, the goal isn't to finish 20 Adventures, but 20 unique Adventures. Since your main character has 18 Adventures completed, even if the two done on your alt are new, if your alt has less than 18 Adventures completed it will still use your main character for the total.