feat: add last_injected_context column to chats table (#23798)

Adds a nullable JSONB column `last_injected_context` to the `chats`
table that stores the most recently persisted injected context parts
(AGENTS.md context-file and skill message parts). The column is updated
only when `persistInstructionFiles()` runs — on first workspace attach
or when the agent changes — so there are no redundant writes on
subsequent turns.

Internal fields (`ContextFileContent`, `ContextFileOS`,
`ContextFileDirectory`, `SkillDir`) are stripped at write time so the
column only holds small metadata. No stripping needed on the read path.

<details>
<summary>Implementation notes</summary>

- New migration `000456` adds nullable `last_injected_context JSONB`
column.
- New SQL query `UpdateChatLastInjectedContext` writes the column
without touching `updated_at`.
- `persistInstructionFiles()` strips internal fields from parts via
`StripInternal()` before persisting.
- Sentinel path (no AGENTS.md) persists skill-only parts when skills
exist.
- `codersdk.Chat` exposes `LastInjectedContext []ChatMessagePart`
(omitempty).
- `db2sdk.Chat()` passes through the already-clean data.

</details>

coderd/database/queries/chats.sql

@@ -528,6 +528,17 @@ WHERE
     id = @id::uuid
 RETURNING *;
 
+-- name: UpdateChatLastInjectedContext :one
+-- Updates the cached injected context parts (AGENTS.md +
+-- skills) on the chat row. Called only when context changes
+-- (first workspace attach or agent change). updated_at is
+-- intentionally not touched to avoid reordering the chat list.
+UPDATE chats SET
+    last_injected_context = sqlc.narg('last_injected_context')::jsonb
+WHERE
+    id = @id::uuid
+RETURNING *;
+
 -- name: UpdateChatMCPServerIDs :one
 UPDATE
     chats