fix(cli): sdk logging is always present even when not turned on (#33324)

### Issue #33320

Closes #33320

### Reason for this change

In #33273 we introduced a bug that causes SDK logs to always be printed.

### Description of changes

Set the correct log level for SDK logs.

### Describe any new or updated permissions being added

n/a

### Description of how you validated changes

Manual test and updated unit test

### Checklist
- [x] My code adheres to the [CONTRIBUTING GUIDE](https://github.com/aws/aws-cdk/blob/main/CONTRIBUTING.md) and [DESIGN GUIDELINES](https://github.com/aws/aws-cdk/blob/main/docs/DESIGN_GUIDELINES.md)

----

*By submitting this pull request, I confirm that my contribution is made under the terms of the Apache-2.0 license*

packages/aws-cdk/lib/api/aws-auth/sdk-logger.ts

@@ -13,7 +13,7 @@ export class SdkToCliLogger implements Logger {
   private notify(level: 'debug' | 'info' | 'warn' | 'error', ...content: any[]) {
     void this.ioHost.notify({
       time: new Date(),
-      level,
+      level: 'trace', // always log all SDK logs at trace level, no matter what level they are coming from the SDK
       action: 'none' as any,
       code: 'CDK_SDK_I0000',
       message: format('[SDK %s] %s', level, formatSdkLoggerContent(content)),
@@ -87,7 +87,7 @@ export class SdkToCliLogger implements Logger {
    * ```
    */
   public error(...content: any[]) {
-    this.notify('info', ...content);
+    this.notify('error', ...content);
   }
 }
 

packages/aws-cdk/test/api/aws-auth/sdk-logger.test.ts

@@ -12,13 +12,16 @@ describe(SdkToCliLogger, () => {
   });
 
   test.each(['trace', 'debug'] as Array<keyof SdkToCliLogger>)('%s method does not call notify', (method) => {
-    logger[method]('test');
+    logger[method]('SDK Logger test message');
     expect(ioHost.notify).not.toHaveBeenCalled();
   });
 
   test.each(['info', 'warn', 'error'] as Array<keyof SdkToCliLogger>)('%s method logs to notify', (method) => {
-    logger[method]('test');
-    expect(ioHost.notify).toHaveBeenCalled();
+    logger[method]('SDK Logger test message');
+    expect(ioHost.notify).toHaveBeenCalledWith(expect.objectContaining({
+      level: 'trace',
+      message: `[SDK ${method}] SDK Logger test message`,
+    }));
   });
 });