docs(phase-9): tighten picker description, swap owner labels

Spec changes from feedback:

1. §5.1 — the picker description was written as if the user could
   see the rendered HTML. Replaced with an ASCII sketch and a
   stronger 'no internal concepts reach the rendered surface' line.

2. §7 — added a binding label-mapping table for every owner-facing
   term. The internal names stay (in the DB, in code, in the spec);
   only rendered text changes. Soft-language variants:
     intake_links     -> 'Request link'
     Mint a link      -> 'New request link'
     Per-link daily cap -> 'Daily cap'
     Revoke           -> 'Pause'
     Re-activate      -> 'Resume'
     show_in_intake   -> 'List on request page'

docs/superpowers/specs/2026-09-21-phase-9-per-client-intake-links-and-project-picker-design.md

@@ -237,7 +237,25 @@ as Phase 8 — it does not care whether the row carries `project` or
 `client`). On unknown/revoked/expired, **404 with the same wording as
 the wizard** so a hostile body cannot probe.
 
-On success, renders a project picker:
+On success, renders a project picker. The user's experience is
+intentionally minimal — they see plain project names, click one, and
+the wizard starts. No mention of phases, tokens, or any other internal
+concept ever reaches the rendered HTML. A rough sketch:
+
+```
+┌─────────────────────────────────────────────┐
+│ Acme — let's talk                           │
+│                                             │
+│ Tell us which project you have a question   │
+│ about:                                      │
+│                                             │
+│   ▸ Acme Site Redesign                      │
+│   ▸ Acme POS                                │
+│   ▸ Acme Rebrand                            │
+└─────────────────────────────────────────────┘
+```
+
+Mechanics:
 
 - Reads every project on the link's client with
   `active = true AND show_in_intake = true`.
@@ -325,24 +343,38 @@ with both tokens and project ids on the wire.
 
 ## 7. Owner UI
 
+The owner-facing surface uses natural-language labels. Internal names
+(`intake_links`, `show_in_intake`) live in the database, in the code,
+and in this spec — never in the rendered HTML. The mapping below is
+binding for every button, header, and tooltip in Phase 9.
+
+| Internal code name | Owner-facing label |
+|---|---|
+| `intake_links` row | **Request link** |
+| "Mint a link" button | **New request link** |
+| "Per-link daily cap" | **Daily cap** |
+| "Revoke" | **Pause** |
+| "Re-activate" | **Resume** |
+| `projects.show_in_intake` toggle | **List on request page** |
+| "Per-project intake count" | "Submissions today" |
+
 ### 7.1 Client admin page (new)
 
-`/admin/clients/[clientId]` currently exists? (No — Phase 8 only added
-links at the project level.) Phase 9 adds:
+`/admin/clients/[clientId]` did not exist before Phase 9 — Phase 8 only
+added request links at the project level. Phase 9 adds:
 
 - A new route `/admin/clients/[clientId]` (server component,
   owner-only via the existing admin layout).
 - Reads the client plus the active projects on it (the owner's view of
   what's deployable for intake purposes).
 - Renders three panels stacked:
-  1. **Public intake links** — the same `<IntakeLinksPanel>` from
+  1. **Request links** — the same `<IntakeLinksPanel>` shape from
      Phase 8, but now `link.project` is always empty on new rows.
-     Mint / rotate / revoke / per-link daily cap.
+     New request link / rotate / pause / resume / daily cap.
   2. **Projects** — a list of the client's projects with a small
-     `<ShowInIntakeToggle>` next to each (one toggle calls one new
-     Server Action per project). Disabled + active styled. Toggling
-     `show_in_intake` is two clicks, not three.
-  3. *(Future)* a "client URL" badge — a copyable URL of the form
+     `<ShowInIntakeToggle>` (rendered as "List on request page") next
+     to each. One toggle calls one new Server Action per project.
+  3. *(Future)* a "Client URL" badge — a copyable URL of the form
      `https://<host>/intake/<token>` — landed on the page when at
      least one active link and at least one opted-in project exist.
 
@@ -355,10 +387,11 @@ needed here.
 
 - A new `<ShowInIntakeToggle projectId={...} current={...} />` rendered
   in the existing admin page, just above the
-  `<BackgroundMdEditor>`. It calls a new Server Action
+  `<BackgroundMdEditor>`. The toggle is labelled "List on request page"
+  in the rendered UI. It calls a new Server Action
   `setProjectShowInIntakeAction({ projectId, show })`. Owner-only.
 
-The Public-intake-links section on the project page stays (Phase 8
+The "Request links" section on the project page stays (Phase 8
 shape: a per-project link can still be minted). It just becomes the
 "legacy / experimental" path; the new client-level link is the
 default.