[Frontend] Standardize API error parsing for JSON, text, and blob responses

## Problem

Different API modules swallow response details or assume one body format, producing misleading failures.

## Relevant code

`src/lib/api`

## Proposed scope

Create one safe parser that preserves status/correlation metadata and never exposes server internals to users.

## Acceptance criteria

- [ ] The reported behavior is reproduced or characterized with a focused automated test before the fix.
- [ ] The implementation satisfies the proposed scope without weakening TypeScript, lint, authorization, privacy, or error handling.
- [ ] Success, empty/loading where applicable, failure, and boundary cases are covered.
- [ ] Existing related tests pass and new regression coverage is included.
- [ ] User-facing behavior remains accessible by keyboard and at mobile viewport sizes where UI is affected.
- [ ] Documentation is updated when the change alters configuration, contracts, security assumptions, or contributor workflow.

## Contributor notes

Base the work on the latest `main` branch. Keep unrelated refactors out of the pull request and include screenshots or recordings for visible changes. Never use real pet, medical, contact, wallet, or credential data in fixtures.

Start in src/lib/api and reproduce or characterize the inconsistent handling with a focused automated test. Trace the API modules that parse responses, then verify the shared behavior across JSON, text, blob, success, empty/loading, failure, and boundary cases while preserving metadata and safe user-facing errors; run related tests and include regression coverage.

Written by the indexing model from the issue text.