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/pi-embedded-runner/run.ts

@@ -274,7 +274,11 @@ export async function runEmbeddedPiAgent(
         params.config,
       );
       if (!model) {
-        throw new Error(error ?? `Unknown model: ${provider}/${modelId}`);
+        throw new FailoverError(error ?? `Unknown model: ${provider}/${modelId}`, {
+          reason: "model_not_found",
+          provider,
+          model: modelId,
+        });
       }
 
       const ctxInfo = resolveContextWindowInfo({