fix(web): handle fulfilled external login errors

[abhinav7x94]

Summary

• centralize Google, GitHub, and AgentID external sign-in handling
• inspect Better Auth's fulfilled result.error contract instead of relying only on rejected promises
• recover from provider and transport failures by clearing the loading state and pending-login markers
• recognize Better Auth's fulfilled status-0 transport errors as network failures
• preserve provider messages for nonzero HTTP errors and keep successful redirect handoffs loading
• add mounted regressions and run the web suite in CI

Why

Better Auth 1.3.3 resolves non-2xx provider failures as { data: null, error }. It also converts fetch rejection into a fulfilled status-0 error. The previous catch-only handlers ignored those results, leaving the login page permanently on “Redirecting…” with stale pending-login state.

Validation

• focused login suite: 8/8
• full web suite: 59/59
• combined queued web tests pass in both file orders
• Next production build
• changed-file TypeScript checks and existing CI type targets
• frozen-lock install, Biome, and git diff --check

Landing note

This branch overlaps the web-test CI/dependency harness in #1498 and #1525. After either lands, rebase this PR and drop the duplicate harness hunks while retaining the login source and regression.