Automatic merge from submit-queue (batch tested with PRs 59873, 59933, 59923, 59944, 59953). If you want to cherry-pick this change to another branch, please follow the instructions <a href="https://github.com/kubernetes/community/blob/master/contributors/devel/cherry-picks.md">here</a>.

Fix DownwardAPI refresh race.

WaitForAttachAndMount should mark only pod in DesiredStateOfWorldPopulator (DSWP) and DSWP should mark the volume to be remounted only when the new pod has been processed.

Otherwise DSWP and reconciler race who gets the new pod first. If it's reconciler, then DownwardAPI and Projected volumes of the pod are not refreshed with new content and they are updated after the next periodic sync (60-90 seconds).

Fixes #59813 

/assign @jingxu97 @saad-ali 
/sig storage
/sig node

```release-note
None
```