Leaderboard score is not your local best score. The highest local score and wave is shown in the level's menu, are you sure you are actually beating that?

I'm not sure at all as I'm quite confused by all those data & local/global state depending on values has been refreshed or not.

I bet it's pretty easy to test on a not yet played level if you got one (or using a test profile or just reset all data for a single level on your test profile) :
-play a level on normal, get a pretty low base score (like 10k)
-wait for the server to refresh (20 mins min right ?)
-reload the game, if it has been closed, the play the level again, the prev score should be 10k, not 0, then beat that (15k for instance)
-wait again
-check which score is displayed on top of the leaderboard : 10 or 15 k ?

As the beta is coming to and end soon, maybe I'll try this myself.

"New high score" labels are not related to the leaderboards at all. They are a local thing and only compare your score to your local progress.
Please try to reproduce it, I'm pretty sure there's no bug here

Still happen, especially with the new season 3, high-scores are clean, giving the opportunity to get the issue.

And as a player, I shouldn't bother if it's a local or online thing, I just saw inconsistent values.

(cannot add pic in this form, so here there is one)
https://i.ibb.co/VSMk8Qx/Infinitode2-perso-record-1.png

Seasons have nothing to do with that - as I've already said, they are stored locally and you have already played that level / have received a higher score before. It worked like that for 5 years.
If you should not bother, then don't.