fix(experiment-tag): use EXP_{slice}_web_exp_id_v2 cookie key

Align root-domain cookie name with the user property name instead of
EXP_{slice}_id_v2.

Co-authored-by: Cursor <cursoragent@cursor.com>

Author: tyiuhc

packages/experiment-tag/src/experiment.ts

@@ -407,7 +407,7 @@ export class DefaultWebExperimentClient implements WebExperimentClient {
       expirationDays: 365,
     });
 
-    const webExpIdV2CookieKey = `${experimentStorageName}_id_v2`;
+    const webExpIdV2CookieKey = `${experimentStorageName}_web_exp_id_v2`;
     const webExpIdV2LocalFallback = user.web_exp_id_v2 ?? user.web_exp_id;
     let generatedSharedId: string | undefined;
     const webExpIdV2 = await resolveCrossSubdomainValue(

packages/experiment-tag/test/experiment.test.ts

@@ -127,7 +127,7 @@ describe('initializeExperiment', () => {
   test('seeds web_exp_id_v2 from existing web_exp_id when cookie is missing', async () => {
     const key = stringify(apiKey);
     const storageKey = 'EXP_' + key;
-    const cookieKey = storageKey + '_id_v2';
+    const cookieKey = storageKey + '_web_exp_id_v2';
     mockGlobal.localStorage.getItem.mockImplementation((name: string) => {
       if (name === storageKey) {
         return JSON.stringify({ web_exp_id: 'existing-id' });