feat: Implement user profile and parent profile button tests

- Added tests for user profile editing, including name changes, image uploads, and password changes.
- Implemented tests for changing the parent PIN with verification code handling.
- Created tests for account deletion with confirmation dialog and email validation.
- Introduced parent profile button tests for both temporary and permanent modes, verifying badge visibility and menu options.
- Updated Playwright configuration to include new test buckets for user profile and parent profile button scenarios.
- Added e2e plans documentation for user profile and parent profile button tests.

frontend/vue-app/e2e/.auth/user.json

@@ -2,17 +2,17 @@
   "cookies": [
     {
       "name": "refresh_token",
-      "value": "13xlv0Lpv2WCL1slwsRXhVfCEBTuzXREnnUYgpfkmRU",
+      "value": "zSbcFMeA59rQCWIkZEDsLO54zjpNu1M0mH0AsHq0Unc",
       "domain": "localhost",
       "path": "/api/auth",
-      "expires": 1781577686.529928,
+      "expires": 1781644571.72276,
       "httpOnly": true,
       "secure": true,
       "sameSite": "Strict"
     },
     {
       "name": "access_token",
-      "value": "eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJlbWFpbCI6ImUyZUB0ZXN0LmNvbSIsInVzZXJfaWQiOiJjYjFhOTYzNy01ZDRkLTRiYTEtODFjNi0xYmM0MjA4M2MyY2YiLCJ0b2tlbl92ZXJzaW9uIjowLCJleHAiOjE3NzM4MDI1ODZ9.jxoB6roEbEMI0NQT3THPC68Q2L-i1gN1KCheJGCUT4U",
+      "value": "eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJlbWFpbCI6ImUyZUB0ZXN0LmNvbSIsInVzZXJfaWQiOiJjMDFmNWJhYS1kMGQ3LTQwYzctOTE2OC1mZGY0M2FiNTVmMzUiLCJ0b2tlbl92ZXJzaW9uIjowLCJleHAiOjE3NzM4Njk0NzF9._LZhytX_HqOpeFQ8XGlHqTz6efhIUDZa2Dp7SB-acqE",
       "domain": "localhost",
       "path": "/",
       "expires": -1,
@@ -27,11 +27,11 @@
       "localStorage": [
         {
           "name": "authSyncEvent",
-          "value": "{\"type\":\"logout\",\"at\":1773801686181}"
+          "value": "{\"type\":\"logout\",\"at\":1773868571358}"
         },
         {
           "name": "parentAuth",
-          "value": "{\"expiresAt\":1773974486751}"
+          "value": "{\"expiresAt\":1774041371935}"
         }
       ]
     }