fix: show the model's replies to background task notifications

Everything the assistant did between a <task-notification> turn and the next
real user message was hidden, so the reply a model gave once a background
command finished (and any tool work after it) never reached the chat, live or
after a reload, even though it was written to the transcript.

Drop that suppression from every session history projection (full, paged,
recovery, sub-agent lookup) and from the desktop store (history mapping and the
live-stream flag). Only the injected notification prompt stays hidden; the
background task cards still come from its notification data. The suppression
was added for #886 to quiet replies to stale notifications; the CLI already
avoids those at the source when the model has read the task's result.

Refs #1389

docs/desktop/sessions.md

@@ -83,7 +83,7 @@ Claude 每完成一轮并修改文件，对话里会出现一张「{n} 个文件
 
 - **任务** — Claude 自己维护的待办清单，顶部显示「任务进度 3/7」。
 - **SubAgent** — 它派出去的子 Agent，点进去能看子 Agent 完整的运行记录。
-- **后台任务** — 挂在后台跑的命令和工作流，可以单独停掉某一个。
+- **后台任务** — 挂在后台跑的命令和工作流，可以单独停掉某一个。任务结束时 Claude 会收到通知并接着处理，它的回复照常出现在对话里。
 - **团队** — 用到 Agent Team 时，每个成员一行，可以点进去直接给某个成员发消息。
 
 后台跑的子 Agent 的工具活动也会冒泡到这里，不用等它跑完才知道它在干什么。

docs/en/desktop/sessions.md

@@ -83,7 +83,7 @@ The first button on the right of the tab bar opens the Activity panel, which lis
 
 - **Tasks** — the to-do list Claude maintains for itself, with "Task progress 3/7" at the top.
 - **SubAgents** — the agents it delegated to. Open one to read its full transcript.
-- **Background tasks** — commands and workflows running in the background; each can be stopped individually.
+- **Background tasks** — commands and workflows running in the background; each can be stopped individually. When one finishes, Claude is notified and carries on, and its reply appears in the conversation as usual.
 - **Team** — when an Agent Team is in play, one row per member, and you can message a member directly.
 
 Tool activity from background subagents bubbles up here too, so you don't have to wait for one to finish to see what it's doing.