Phase 1: Refactor all 16 element component interfaces to use MCP SDK types

Replace ad-hoc local type definitions in element components with real
MCP SDK types and Inspector wrapper types from core/mcp/types.ts:

Schema-typed primitives:
- LogLevelBadge: LogLevel → LoggingLevel (SDK)
- TransportBadge: "stdio"|"http" → ServerType (core), add sse/streamable-http
- TaskStatusBadge: local TaskStatus → SDK TaskStatus (working/input_required/...)
- ServerStatusIndicator: inline union → ConnectionStatus (core), "failed"→"error"

MCP wrapper consumers:
- AnnotationBadge: label+variant → facet+value with SDK Role type
- CapabilityItem: name:string → capability:keyof ServerCapabilities|ClientCapabilities
- LogEntry: flat props → entry:{receivedAt:Date, params:LoggingMessageNotification["params"]}
- MessageBubble: flat props → message:SamplingMessage|PromptMessage
- ProgressDisplay: flat props → params:ProgressNotification["params"]

Composite-input elements:
- ContentViewer: type+content+mimeType → block:ContentBlock (SDK discriminated union)
- InlineError: message+details → error:{message,data?}
- ConnectionToggle: checked+loading+onChange → status:ConnectionStatus+onConnect/onDisconnect

Also updates all group/screen/view consumers to keep the build green,
and renames CSS variable --inspector-status-failed → --inspector-status-error.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

clients/web/src/components/elements/TransportBadge/TransportBadge.stories.tsx

@@ -15,8 +15,14 @@ export const Stdio: Story = {
   },
 };
 
-export const Http: Story = {
+export const Sse: Story = {
   args: {
-    transport: "http",
+    transport: "sse",
+  },
+};
+
+export const StreamableHttp: Story = {
+  args: {
+    transport: "streamable-http",
   },
 };

clients/web/src/components/elements/TransportBadge/TransportBadge.tsx

@@ -1,13 +1,20 @@
 import { Badge } from "@mantine/core";
+import type { ServerType } from "@inspector/core/mcp/types.js";
 
 export interface TransportBadgeProps {
-  transport: "stdio" | "http";
+  transport: ServerType;
 }
 
+const transportLabel: Record<ServerType, string> = {
+  stdio: "STDIO",
+  sse: "SSE",
+  "streamable-http": "HTTP",
+};
+
 export function TransportBadge({ transport }: TransportBadgeProps) {
   return (
     <Badge variant="outline" color="gray">
-      {transport.toUpperCase()}
+      {transportLabel[transport]}
     </Badge>
   );
 }