fix(auth): bidirectional mode/type compat + sync OAuth to all agents (#12692)

Merged via /review-pr -> /prepare-pr -> /merge-pr.

Prepared head SHA: 2dee8e1174
Co-authored-by: mudrii <220262+mudrii@users.noreply.github.com>
Co-authored-by: obviyus <22031114+obviyus@users.noreply.github.com>
Reviewed-by: @obviyus

src/agents/model-fallback.e2e.test.ts

@@ -262,6 +262,49 @@ describe("runWithModelFallback", () => {
     ]);
   });
 
+  it("falls back on unknown model errors", async () => {
+    const cfg = makeCfg();
+    const run = vi
+      .fn()
+      .mockRejectedValueOnce(new Error("Unknown model: anthropic/claude-opus-4-6"))
+      .mockResolvedValueOnce("ok");
+
+    const result = await runWithModelFallback({
+      cfg,
+      provider: "anthropic",
+      model: "claude-opus-4-6",
+      run,
+    });
+
+    // Override model failed with model_not_found → falls back to configured primary.
+    // (Same candidate-resolution path as other override-model failures.)
+    expect(result.result).toBe("ok");
+    expect(run).toHaveBeenCalledTimes(2);
+    expect(run.mock.calls[1]?.[0]).toBe("openai");
+    expect(run.mock.calls[1]?.[1]).toBe("gpt-4.1-mini");
+  });
+
+  it("falls back on model not found errors", async () => {
+    const cfg = makeCfg();
+    const run = vi
+      .fn()
+      .mockRejectedValueOnce(new Error("Model not found: openai/gpt-6"))
+      .mockResolvedValueOnce("ok");
+
+    const result = await runWithModelFallback({
+      cfg,
+      provider: "openai",
+      model: "gpt-6",
+      run,
+    });
+
+    // Override model failed with model_not_found → falls back to configured primary.
+    expect(result.result).toBe("ok");
+    expect(run).toHaveBeenCalledTimes(2);
+    expect(run.mock.calls[1]?.[0]).toBe("openai");
+    expect(run.mock.calls[1]?.[1]).toBe("gpt-4.1-mini");
+  });
+
   it("skips providers when all profiles are in cooldown", async () => {
     const provider = `cooldown-test-${crypto.randomUUID()}`;
     const profileId = `${provider}:default`;