[v13] Move lib/web tests to new web login helpers (#38890)

* Move lib/web tests to new web login helpers (#38806)

* Move login helpers to a separate file

* Reuse newOTPSharedSecret

* Add login method variants and tweak signatures

* Fix TestWebSessionsBadInput

* Move tests to new web login helpers

* Unify and simplify helpers, verify response status

* Fix TestPasswordChange

* Drop unused functions from backport