fix(agent): inspect wrapped API error bodies

[sgaofen]

Summary

• Fixes Wrapped API errors lose nested body details and misclassify transient 402s as billing failures #14195.
• Makes _extract_error_body() walk wrapped exception cause/context chains, matching _extract_status_code().
• Preserves top-level body precedence while allowing nested SDK bodies to drive retry/billing classification.

Root cause

Wrapped SDK exceptions could expose a nested HTTP status code while hiding the nested structured body. That made transient 402 usage-limit responses look like plain billing failures.

Tests

• uv run --frozen --python 3.11 --extra dev pytest -o addopts='' tests/agent/test_error_classifier.py -q
• git diff --check

[teknium1]

Fixed via PR #53582 (#53582), which salvaged @LeonSGP43's #14287 onto current main. Your PR fixed the same issue (#14195) with an equivalent approach — thanks for the contribution and the duplicate coverage.