Address review on MediaSourceCount batching

Rename GetItemsWithAlternateVersions to GetItemIdsWithAlternateVersions
across the interfaces and implementations since it returns ids. Return
the hashset straight from the query instead of materializing an array
first. Rename the DtoService guard to mayHaveAlternateVersions and
invert it so the computed path is the explicit case. Assert the media
source count value in the batch skip test and add a test covering an
item that is in the returned set still resolving to the correct count.

Emby.Server.Implementations/Dto/DtoService.cs

@@ -261,7 +261,7 @@ namespace Emby.Server.Implementations.Dto
                 var versionItemIds = accessibleItems.OfType<Video>().Select(i => i.Id).ToList();
                 if (versionItemIds.Count > 0)
                 {
-                    alternateVersionItemIds = _libraryManager.GetItemsWithAlternateVersions(versionItemIds);
+                    alternateVersionItemIds = _libraryManager.GetItemIdsWithAlternateVersions(versionItemIds);
                 }
             }
 
@@ -1314,13 +1314,15 @@ namespace Emby.Server.Implementations.Dto
                 if (options.ContainsField(ItemFields.MediaSourceCount))
                 {
                     // A video with no primary version and no alternate versions always has a single
-                    // media source. When the batch has already determined this item owns no alternate
-                    // versions, skip the per-item alternate-version queries entirely (the common case).
-                    var hasNoAlternateVersions = alternateVersionItemIds is not null
-                        && !video.PrimaryVersionId.HasValue
-                        && !alternateVersionItemIds.Contains(video.Id);
+                    // media source. Only compute the count for videos that might have more: a primary
+                    // version, or membership in the batch's set of items that own alternate versions.
+                    // Without the batch we can't rule it out, so fall back to computing (the single-item
+                    // path). Everything else is the common case and keeps the default count of one.
+                    var mayHaveAlternateVersions = alternateVersionItemIds is null
+                        || video.PrimaryVersionId.HasValue
+                        || alternateVersionItemIds.Contains(video.Id);
 
-                    if (!hasNoAlternateVersions)
+                    if (mayHaveAlternateVersions)
                     {
                         // Match the per-user filtering of the media sources: versions the user cannot
                         // access are not selectable, so they must not count towards the badge either.

Emby.Server.Implementations/Library/LibraryManager.cs

@@ -2235,9 +2235,9 @@ namespace Emby.Server.Implementations.Library
         }
 
         /// <inheritdoc />
-        public IReadOnlySet<Guid> GetItemsWithAlternateVersions(IReadOnlyList<Guid> itemIds)
+        public IReadOnlySet<Guid> GetItemIdsWithAlternateVersions(IReadOnlyList<Guid> itemIds)
         {
-            return _linkedChildrenService.GetItemsWithAlternateVersions(itemIds);
+            return _linkedChildrenService.GetItemIdsWithAlternateVersions(itemIds);
         }
 
         /// <inheritdoc />