Force to refresh sync cycle(poll/nudge) when transiting from CONFIGURATION to NORMAL mode. (uplift to 1.13.x)

[brave-builds]

Uplift of #6363
Resolves brave/brave-browser#11116

Approved, please ensure that before merging:

• You have checked CI and the builds, lint, and tests all pass or are not related to your PR.
• You have tested your change on Nightly.
• The PR milestones match the branch they are landing to.

After you merge:

• The associated issue milestone is set to the smallest version that the changes is landed on.

[kjozwiak]

test-unit failed on Win x64 due to GreaselionServiceTest.ScriptInjectionWithPrecondition as per https://ci.brave.com/job/pr-brave-browser-pr6363_sync-v2-initial-merge_1.13.x/1/execution/node/676/log/. I'm not 100% sure if this is an intermittent issue but we've seen a few times before via the CR85 branch. Going to restart just in case.

20:31:50  [  FAILED  ] GreaselionServiceTest.ScriptInjectionWithPrecondition, where TypeParam =  and GetParam() =  (2669 ms)
20:31:50  
20:31:50  [600/602] GreaselionServiceTest.ScriptInjectionWithPrecondition (2840 ms)
20:31:50  [601/602] BraveAdsBrowserTest/BraveAdsUpgradeBrowserTest.PRE_UpgradePath/PreferencesForVersion068WithRewardsAndAdsEnabled_ForUnsupportedLocale_ForUnsupportedLocale_RewardsShouldBeEnabled_AdsShouldBeDisabled_ShouldNotShowOnboarding (483 ms)
20:31:50  [602/602] BraveAdsBrowserTest/BraveAdsUpgradeBrowserTest.UpgradePath/PreferencesForVersion068WithRewardsAndAdsEnabled_ForUnsupportedLocale_ForUnsupportedLocale_RewardsShouldBeEnabled_AdsShouldBeDisabled_ShouldNotShowOnboarding (353 ms)
20:31:50  1 test failed:
20:31:50      GreaselionServiceTest.ScriptInjectionWithPrecondition (../../brave/browser/greaselion/greaselion_browsertest.cc:263)

[kjozwiak]

Restarting Win x64 again as we lost the spot instance/agent was removed as per https://ci.brave.com/job/pr-brave-browser-pr6363_sync-v2-initial-merge_1.13.x/2/execution/node/375/log/

01:54:51  Cannot contact i-0d57044b16f543d42: hudson.remoting.RequestAbortedException: java.io.IOException: Unexpected termination of the channel
01:59:57  Could not connect to i-0d57044b16f543d42 to send interrupt signal to process

[kjozwiak]

@darkdh you'll most likely need to rebase this one as #6276 & brave/brave-browser#11026 landed in 1.13.x and 1.12.x which I believe is the reason the above is now failing init due to the following:

02:54:56  1 failed patches:
02:54:56  chrome/VERSION
02:54:56    - Patch applied because: No corresponding .patchinfo file was found.
02:54:56    - Error - Program git exited with error code 1.

[kjozwiak]

Uplift into 1.13.x approved after deliberating with @brave/uplift-approvers. QA has also verified the PR on Nightly as per #6363 (comment).