feat(analytics): E-03 track the save-login prompt and login results

[navidshad]

What

New and changed events for the "Login to save this phrase" path:

Event	When	Properties
save-login-prompt_viewed	The button shows on the word card. Once per word view (a new word or a new view resets it).	—
save-login-prompt_clicked	The button is clicked.	—
user_logged-in	A login from the login window works.	new: source
login_failed (new)	A login from the login window fails.	source, reason

• source: save_prompt (word card), practice (practice page), popup (toolbar popup, or no source).
• reason:
  • window_closed: the user closed the Google consent window.
  • google_error: Google gave no token (also the "Current Chrome User" button when Chrome has no token).
  • server_error: our server failed or sent no token, or the email/password login was rejected.
• practice-now_login-clicked is kept.

How the source reaches the login window

The login runs in a separate window (popup.html#/login), opened by the background. So OpenLoginWindowMessage now carries an optional source. The background adds it to the URL (#/login?source=save_prompt). LoginView reads it from the route.

Why

94% of people who look up a word are not logged in. They see this button instead of Save, but we could not tell if they ignore it, click it, or click it and fail. Work list: subturtle-doc docs/tech/analytics-fixes-implementation.md (row E-03). The funnel viewed → clicked → user_logged-in (source save_prompt) → phrase_saved now works in Mixpanel.

Small fix on the way

In the Google flow, a server reply with no token used to be stored as a login (an empty token). Now it counts as login_failed with server_error, and nothing is stored.

How tested

• New unit tests: tests/save-login-prompt.test.ts (viewed once, again for a new word, not for a logged-in user, clicked + source) and tests/login-analytics.test.ts (helpers, source on user_logged-in, each failure reason).
• yarn typecheck: clean.
• yarn test: 42 files, 370 tests pass.
• yarn build: OK.
• npx playwright test tests/e2e/password-login.spec.ts: 4 passed (this spec drives LoginView).
• The repo has no lint script.

Dependencies

• None to merge. It works best with fix(analytics): E-01 E-05 send the install id and trigger with translation calls #58 (E-01), so these events and the server events share one install id.

🤖 Generated with Claude Code

[navidshad]

Replaced by #62, which combines all extension rows (E-01 to E-06) into one PR.