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/mode_parent/profile-button/profile-button-temporary.spec.ts

@@ -0,0 +1,25 @@
+import { test, expect } from '@playwright/test'
+import { E2E_PIN } from '../../e2e-constants'
+
+test.describe('Parent profile button – temporary parent mode', () => {
+  test.beforeEach(async ({ page }) => {
+    await page.goto('/parent')
+    // Switch from permanent mode to temporary mode:
+    // 1. Exit parent mode via Child Mode
+    await page.getByRole('button', { name: 'Parent menu' }).click()
+    await page.getByRole('menuitem', { name: 'Child Mode' }).click()
+    await expect(page.getByRole('button', { name: 'Parent login' })).toBeVisible()
+    // 2. Re-enter parent mode WITHOUT the persistence checkbox
+    await page.getByRole('button', { name: 'Parent login' }).click()
+    const pinInput = page.getByPlaceholder('4–6 digits')
+    await pinInput.waitFor({ timeout: 5000 })
+    await pinInput.fill(E2E_PIN)
+    // Do NOT check 'Stay in parent mode'
+    await page.getByRole('button', { name: 'OK' }).click()
+    await page.waitForURL(/\/parent(\/|$)/)
+  })
+
+  test('Badge – no 🔒 badge in temporary parent mode', async ({ page }) => {
+    await expect(page.getByLabel('Persistent parent mode active')).not.toBeVisible()
+  })
+})