Informed error parsing from api (#1327)

## Why:
Errors were being wrapped under an opaque stagehand server error. This
PR helps provide clearer, actionable errors.

## What:
- On system events with status=error, throw new Error with the server
message.
- In the catch block, rethrow non-SyntaxError Errors; only wrap
JSON.parse SyntaxError as StagehandResponseParseError.

## Test Plan:
- [x] Unit test: status=error yields a plain Error with the raw server
message.
- [x] Unit test: invalid JSON is wrapped as StagehandResponseParseError.

Author: miguelg719

.changeset/kind-snails-judge.md

@@ -0,0 +1,5 @@
+---
+"@browserbasehq/stagehand": patch
+---
+
+Informed error parsing from api

packages/core/lib/v3/api.ts

@@ -415,17 +415,25 @@ export class StagehandAPIClient {
 
           if (eventData.type === "system") {
             if (eventData.data.status === "error") {
-              throw new StagehandServerError(eventData.data.error);
+              const { error: errorMsg } = eventData.data;
+              // Throw plain Error to match local SDK behavior (useApi: false)
+              throw new Error(errorMsg);
             }
             if (eventData.data.status === "finished") {
               return eventData.data.result as T;
             }
           } else if (eventData.type === "log") {
+            const msg = eventData.data.message;
+            // Skip server-side internal logs that don't apply to API mode
+            if (msg?.message === "Connecting to local browser") {
+              continue;
+            }
             this.logger(eventData.data.message);
           }
         } catch (e) {
-          // Don't catch and re-throw StagehandServerError
-          if (e instanceof StagehandServerError) {
+          // Let Error instances pass through (server errors thrown above)
+          // Only wrap SyntaxError from JSON.parse as parse errors
+          if (e instanceof Error && !(e instanceof SyntaxError)) {
             throw e;
           }