I used this simple code to set a Notification in Android 4.1 or higher.
It works well, but my problem comes with SmallIcon and LargeIcon.
I understand that SmallIcon is shown in the status bar and the LargeIcon is shown in the dropdown list.
NotificationCompat.Builder builder = new NotificationCompat.Builder(this);
builder.setTicker("The ticker");
builder.setContentTitle("The title");
builder.setContentText("The text");
builder.setSmallIcon(R.drawable.my_small_icon);
Bitmap bm = BitmapFactory.decodeResource(getResources(), R.drawable.my_96px_large_icon);
builder.setLargeIcon(bm);
NotificationManager nm = (NotificationManager) getSystemService(NOTIFICATION_SERVICE);
nm.notify("direct_tag", NOTIF_ALERTA_ID, builder.build());
My problem is:
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When the notification is launched, a cropped oversized Small Icon is shown next to “The Ticker” text, instead of showing the original SmallIcon without oversizing it.

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In the dropdown list I see the LargeIcon on the left, that’s good. But I also see the small icon on the right, next to the time of the notification. I don’t want to show that.

In my application I provide large (128×128 px) PNG drawable as small icon, and it shows scaled and without cropping. Is your drawable defined in bitmap file or maybe as XML resource? In XML you can specify several aspects of display (e.g. cropping). Double check your XML or use just PNG/JPG.
As Android API documentation on Notification.setSmallIcon() clearly states:
AFAIK there’s no way you can override the behaviour, unless you provide your own notification template (via Notification.setContent()