[ie/youtube] Fix `web_embedded` client extraction (#14843)

Closes #14826
Authored by: bashonly, seproDev

Co-authored-by: sepro <sepro@sepr0.com>

yt_dlp/extractor/youtube/_base.py

@@ -380,10 +380,13 @@ def short_client_name(client_name):
     return join_nonempty(main[:4], ''.join(x[0] for x in parts)).upper()
 
 
-def build_innertube_clients():
+def _fix_embedded_ytcfg(ytcfg):
-    THIRD_PARTY = {
+    ytcfg['INNERTUBE_CONTEXT'].setdefault('thirdParty', {}).update({
         'embedUrl': 'https://www.youtube.com/',  # Can be any valid URL
-    }
+    })
+
+
+def build_innertube_clients():
     BASE_CLIENTS = ('ios', 'web', 'tv', 'mweb', 'android')
     priority = qualities(BASE_CLIENTS[::-1])
 
@@ -405,7 +408,7 @@ def build_innertube_clients():
         ytcfg['priority'] = 10 * priority(base_client)
 
         if variant == 'embedded':
-            ytcfg['INNERTUBE_CONTEXT']['thirdParty'] = THIRD_PARTY
+            _fix_embedded_ytcfg(ytcfg)
             ytcfg['priority'] -= 2
         elif variant:
             ytcfg['priority'] -= 3
@@ -991,6 +994,10 @@ class YoutubeBaseInfoExtractor(InfoExtractor):
 
         ytcfg = self.extract_ytcfg(video_id, webpage) or {}
 
+        # See https://github.com/yt-dlp/yt-dlp/issues/14826
+        if _split_innertube_client(client)[2] == 'embedded':
+            _fix_embedded_ytcfg(ytcfg)
+
         # Workaround for https://github.com/yt-dlp/yt-dlp/issues/12563
         # But it's not effective when logged-in
         if client == 'tv' and not self.is_authenticated: