To make it more clear I'll call the first activity (the one with the list) - ListActivity, and the new activity (the one with the buttons) - ButtonActivity.
To detect the button that was pressed in ButtonActivity add an implement to the class like this:
public class ButtonActivity extends AppCompatActivity implements View.OnClickListener
Then implement the method onClick(View v), like this :
@Override
public void onClick(View v) {
    int id = v.getId();
    switch (id) {
        case R.id.your_save_button_id: {
            Intent data = new Intent();
            data.putExtra("item_to_save", item);
            setResult(CommonStatusCodes.SUCCESS, data);
            finish();
            break;
        }
        case R.id.your_delete_button_id: {
            Intent data = new Intent();
            data.putExtra("item_to_delete", item);
            setResult(CommonStatusCodes.SUCCESS, data);
            finish();
            break;
        }
        default: {
            break;
        }
    }
}
In your ListActivity:
@Override
protected void onActivityResult(int requestCode, int resultCode, Intent data) {
    if (requestCode == "request_code_for_button_activity") {
        if (resultCode == CommonStatusCodes.SUCCESS) {
            if (data != null) {
                if (data.getStringExtra("item_to_delete") != null) {
                    //DELETE YOUR ITEM HERE
                } else if (data.getStringExtra("item_to_save") != null) {
                    //SAVE YOUR ITEM HERE
                }
            }
        }
    } else {
        super.onActivityResult(requestCode, resultCode, data);
    }
}