Address PR #197 review: test coverage + dedup + CI log capture

Code-review follow-ups (Approve-with-comments) for batch #197
(context badge #189 / e2e in CI #187 / inline MCP test #170):

- server: extract the duplicated chatContextWindow ::text->positive-int
  coercion (resolve() + getMasked()) into an exported parsePositiveInt
  helper and unit-test its branches (200000/1.9/0/-5/""/abc/undefined),
  closing the untested read-path gap.
- client: merge the two backward scans over messageRows into one pure,
  exported selectContextBadge helper (numerator and denominator still
  taken from the most recent row carrying EACH value) and unit-test the
  different-rows and fresh-zero-doesn't-shadow cases.
- client: extract the MCP "Test" button tristate presentation into a pure
  mcpTestButtonView helper (collapses the two parallel if/else chains) and
  unit-test idle/ok-with-tools/ok-no-tools/failed label+tooltip branches.
- ci: redirect the backgrounded prod server's stdout/stderr to a log file
  in e2e-mcp and cat it on failure, so a start-up crash is diagnosable
  instead of surfacing only as the generic health timeout.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

.github/workflows/develop.yml

@@ -180,7 +180,10 @@ jobs:
         run: pnpm --filter ./apps/server migration:latest
 
       - name: Start server (prod)
-        run: pnpm --filter ./apps/server start:prod &
+        # Capture stdout/stderr so a start-up crash (bind error, stack trace,
+        # migration mismatch) is diagnosable; without this the only signal is
+        # the generic health-loop timeout below, ~120s later.
+        run: pnpm --filter ./apps/server start:prod > /tmp/server.log 2>&1 &
 
       - name: Wait for server health
         run: |
@@ -194,6 +197,10 @@ jobs:
           echo "Server did not become healthy in time"
           exit 1
 
+      - name: Dump server log on failure
+        if: failure()
+        run: cat /tmp/server.log || true
+
       - name: Seed admin
         run: |
           curl -fsS -X POST http://localhost:3000/api/auth/setup \