refactor(provider): switch from full config overwrite to partial key-field merging

Replace the provider switching mechanism for Claude/Codex/Gemini from
full settings_config overwrite to partial key-field replacement, preserving
user's non-provider settings (plugins, MCP, permissions, etc.) across switches.

- Add write_live_partial() with per-app implementations for Claude (JSON env
  merge), Codex (auth replace + TOML partial merge), and Gemini (env merge)
- Add backfill_key_fields() to extract only provider-specific fields when
  saving live config back to provider entries
- Update switch_normal, sync_current_to_live, add, update to use partial merge
- Remove common config snippet feature for Claude/Codex/Gemini (no longer
  needed with partial merging); preserve OMO common config
- Delete 6 frontend files (3 components + 3 hooks), clean up 11 modified files
- Remove backend extract_common_config_* methods, 3 Tauri commands,
  CommonConfigSnippets struct, and related migration code
- Update integration tests to validate key-field-only backfill behavior

src-tauri/src/database/tests.rs

@@ -513,8 +513,6 @@ fn schema_dry_run_does_not_write_to_disk() {
         mcp: Default::default(),
         prompts: Default::default(),
         skills: Default::default(),
-        common_config_snippets: Default::default(),
-        claude_common_config_snippet: None,
     };
 
     // Dry-run should succeed without any file I/O errors
@@ -563,8 +561,6 @@ fn dry_run_validates_schema_compatibility() {
         mcp: Default::default(),
         prompts: Default::default(),
         skills: Default::default(),
-        common_config_snippets: Default::default(),
-        claude_common_config_snippet: None,
     };
 
     // Dry-run should validate the full migration path