diff --git a/yt_dlp/extractor/common.py b/yt_dlp/extractor/common.py
index 39c855c4e0..b4f7250f7f 100644
--- a/yt_dlp/extractor/common.py
+++ b/yt_dlp/extractor/common.py
@@ -877,13 +877,8 @@ class InfoExtractor:
         else:
             self._downloader._first_webpage_request = False
 
-        if note is None:
-            self.report_download_webpage(video_id)
-        elif note is not False:
-            if video_id is None:
-                self.to_screen(str(note))
-            else:
-                self.to_screen(f'{video_id}: {note}')
+        if note is not False:
+            self.report_download_webpage(video_id, note=note)
 
         # Some sites check X-Forwarded-For HTTP header in order to figure out
         # the origin of the client behind proxy. This allows bypassing geo
@@ -1073,9 +1068,9 @@ class InfoExtractor:
 
     def __print_error(self, errnote, fatal, video_id, err):
         if fatal:
-            raise ExtractorError(f'{video_id}: {errnote}', cause=err)
+            raise ExtractorError(f'{errnote}', cause=err, video_id=video_id)
         elif errnote:
-            self.report_warning(f'{video_id}: {errnote}: {err}')
+            self.report_warning(f'{errnote}: {err}', video_id=video_id)
 
     def _parse_xml(self, xml_string, video_id, transform_source=None, fatal=True, errnote=None):
         if transform_source:
@@ -1236,9 +1231,9 @@ class InfoExtractor:
         """Report information extraction."""
         self.to_screen(f'{id_or_name}: Extracting information')
 
-    def report_download_webpage(self, video_id):
+    def report_download_webpage(self, video_id, *, note=None):
         """Report webpage download."""
-        self.to_screen(f'{video_id}: Downloading webpage')
+        self.to_screen(join_nonempty(video_id, note or 'Downloading webpage', delim=': '))
 
     def report_age_confirmation(self):
         """Report attempt to confirm age."""