Update the implementation to ensure it matches the latest design (#6522)

Task/Issue URL:
https://app.asana.com/1/137249556945/project/1200581511062568/task/1210823544400071?focus=true

### Description
This commit refactors the Set as Default Browser message interactions,
UI and associated pixel events. Key changes include:

- Added a "Do Not Ask Again" button to replace the "Not Now" button in
the default browser dialog
- Updated the dialog UI with a close button and improved styling
- Renamed methods and pixels to reflect the new "Do Not Ask Again"
functionality
- Improved user type detection (new vs existing users) with different
prompt flows for each
- Simplified stage progression logic with separate paths for new and
existing users
- Updated menu item styling and icons to match design guidelines
- Fixed message visibility logic to prevent showing low priority
messages when remote messages are active

### Steps to test this PR
Code review only.

### UI changes
| Updated Menu Item  | Updated prompt |
| ------ | ----- |
|<img width="1080" height="2400" alt="updated_menu_item_portrait"
src="https://github.com/user-attachments/assets/5a8c7bc0-e839-4d13-a4cc-0190fc54e624"
/>|<img width="1080" height="2400" alt="prompt_on_tab_showing_website"
src="https://github.com/user-attachments/assets/d714f211-d162-469d-bf56-cbd8dbbf3aeb"
/>|

Author: anikiki

app/src/main/java/com/duckduckgo/app/browser/BrowserActivity.kt

@@ -712,6 +712,7 @@ open class BrowserActivity : DuckDuckGoActivity() {
             is Command.OpenSavedSite -> currentTab?.openSavedSite(command.url)
             is Command.ShowSetAsDefaultBrowserDialog -> showSetAsDefaultBrowserDialog()
             is Command.DismissSetAsDefaultBrowserDialog -> dismissSetAsDefaultBrowserDialog()
+            is Command.DoNotAskAgainSetAsDefaultBrowserDialog -> dismissSetAsDefaultBrowserDialog()
             is Command.ShowSystemDefaultAppsActivity -> showSystemDefaultAppsActivity(command.intent)
             is Command.ShowSystemDefaultBrowserDialog -> showSystemDefaultBrowserDialog(command.intent)
             is Command.ShowUndoDeleteTabsMessage -> showTabsDeletedSnackbar(command.tabIds)
@@ -1262,8 +1263,8 @@ open class BrowserActivity : DuckDuckGoActivity() {
                 viewModel.onSetDefaultBrowserConfirmationButtonClicked()
             }
 
-            override fun onNotNowButtonClicked() {
-                viewModel.onSetDefaultBrowserNotNowButtonClicked()
+            override fun onDoNotAskAgainButtonClicked() {
+                viewModel.onSetDefaultBrowserDoNotAskAgainButtonClicked()
             }
         }
         dialog.show()

app/src/main/java/com/duckduckgo/app/browser/BrowserViewModel.kt

@@ -25,6 +25,7 @@ import androidx.lifecycle.viewModelScope
 import com.duckduckgo.anvil.annotations.ContributesRemoteFeature
 import com.duckduckgo.anvil.annotations.ContributesViewModel
 import com.duckduckgo.app.browser.BrowserViewModel.Command.DismissSetAsDefaultBrowserDialog
+import com.duckduckgo.app.browser.BrowserViewModel.Command.DoNotAskAgainSetAsDefaultBrowserDialog
 import com.duckduckgo.app.browser.BrowserViewModel.Command.LaunchTabSwitcher
 import com.duckduckgo.app.browser.BrowserViewModel.Command.OpenDuckChat
 import com.duckduckgo.app.browser.BrowserViewModel.Command.ShowUndoDeleteTabsMessage
@@ -129,6 +130,7 @@ class BrowserViewModel @Inject constructor(
         data class OpenSavedSite(val url: String) : Command()
         data object ShowSetAsDefaultBrowserDialog : Command()
         data object DismissSetAsDefaultBrowserDialog : Command()
+        data object DoNotAskAgainSetAsDefaultBrowserDialog : Command()
         data class ShowSystemDefaultBrowserDialog(val intent: Intent) : Command()
         data class ShowSystemDefaultAppsActivity(val intent: Intent) : Command()
         data class ShowUndoDeleteTabsMessage(val tabIds: List<String>) : Command()
@@ -399,6 +401,7 @@ class BrowserViewModel @Inject constructor(
     }
 
     fun onSetDefaultBrowserDialogCanceled() {
+        command.value = DismissSetAsDefaultBrowserDialog
         additionalDefaultBrowserPrompts.onMessageDialogCanceled()
     }
 
@@ -407,9 +410,9 @@ class BrowserViewModel @Inject constructor(
         additionalDefaultBrowserPrompts.onMessageDialogConfirmationButtonClicked()
     }
 
-    fun onSetDefaultBrowserNotNowButtonClicked() {
-        command.value = DismissSetAsDefaultBrowserDialog
-        additionalDefaultBrowserPrompts.onMessageDialogNotNowButtonClicked()
+    fun onSetDefaultBrowserDoNotAskAgainButtonClicked() {
+        command.value = DoNotAskAgainSetAsDefaultBrowserDialog
+        additionalDefaultBrowserPrompts.onMessageDialogDoNotAskAgainButtonClicked()
     }
 
     fun onSystemDefaultBrowserDialogShown() {

app/src/main/java/com/duckduckgo/app/browser/defaultbrowsing/prompts/AdditionalDefaultBrowserPrompts.kt

@@ -33,14 +33,16 @@ interface AdditionalDefaultBrowserPrompts {
     fun onMessageDialogShown()
     fun onMessageDialogCanceled()
     fun onMessageDialogConfirmationButtonClicked()
-    fun onMessageDialogNotNowButtonClicked()
+    fun onMessageDialogDoNotAskAgainButtonClicked()
 
     fun onSystemDefaultBrowserDialogShown()
     fun onSystemDefaultBrowserDialogSuccess(trigger: SetAsDefaultActionTrigger)
     fun onSystemDefaultBrowserDialogCanceled(trigger: SetAsDefaultActionTrigger)
 
     fun onSystemDefaultAppsActivityClosed(trigger: SetAsDefaultActionTrigger)
 
+    fun onUserMessageInteraction(doNotShowAgain: Boolean = false)
+
     sealed class Command {
         data object OpenMessageDialog : Command()
         data class OpenSystemDefaultBrowserDialog(
@@ -55,8 +57,9 @@ interface AdditionalDefaultBrowserPrompts {
     }
 
     enum class SetAsDefaultActionTrigger {
-        DIALOG,
+        PROMPT,
         MENU,
+        MESSAGE,
         UNKNOWN,
     }
 }