Split release publishing into a separate public distribution repo

Gitea has no per-release visibility override -- a repo is public or
private for everything (raw files, releases, API), confirmed against
Gitea's own docs/issue tracker. So manifest.json and release zips now
publish to a dedicated public repo (JellyfinSyncPlus-repo) via Gitea's
Contents API, while this source repo stays private. The v0.2.0.0 release
already pushed under the old (source-repo-only) version of this pipeline
is now stale; the next tag will publish correctly under the new split.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

manifest.json

@@ -1,20 +0,0 @@
-[
-    {
-        "guid": "267dbfe9-bb9c-4eeb-97aa-f0449283cfe6",
-        "name": "JellyfinSyncPlus",
-        "description": "Read-only live view of every active SyncPlay group's members -- drift, position, play method, bitrate, and codec info -- plus a manual \"sync me to group\" button for one-shot self-correction. Does not touch playback automatically: earlier attempts at automatic quality-forcing and automatic drift correction both caused real problems in live use and were reverted (see PLAN.md). Reachable via Dashboard -> Plugins -> JellyfinSyncPlus.",
-        "overview": "SyncPlay stats for nerds: live drift/playback view plus a manual one-shot sync button.",
-        "owner": "cynic",
-        "category": "General",
-        "versions": [
-            {
-                "version": "0.2.0.0",
-                "changelog": "0.2.0.0: SyncPlay stats-for-nerds page and manual one-shot \"sync me to group\" button. No automatic playback intervention. 0.1.0.0: Phase 0 scaffold. Trivial plugin, no functionality yet.",
-                "targetAbi": "10.11.6.0",
-                "sourceUrl": "https://gitea.mrcynic.site/seer/JellyfinSyncPlus/releases/download/v0.2.0.0/JellyfinSyncPlus_0.2.0.0.zip",
-                "checksum": "34e384fc5914e2c459f52a9a7bfd91a5",
-                "timestamp": "2026-07-09T15:11:18Z"
-            }
-        ]
-    }
-]