fix: align Implementation.Name with VS Code config key

Set the protocol-level Implementation.Name to "github" so the slug
matches the configuration key VS Code derives from server.json's title.
This fixes "Unknown tool 'github/github-mcp-server/*'" when references
use the recommended "github/*" pattern.

The MCP registry name in server.json
(io.github.github/github-mcp-server) intentionally still uses the
source-repo's last segment per the registry naming convention; only the
client-facing Implementation.Name changes here.

Fixes #1402

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>

Author: yashhzd

pkg/github/server.go

@@ -87,7 +87,7 @@ func NewMCPServer(ctx context.Context, cfg *MCPServerConfig, deps ToolDependenci
 		o(serverOpts)
 	}
 
-	ghServer := NewServer(cfg.Version, cfg.Translator("SERVER_NAME", "github-mcp-server"), cfg.Translator("SERVER_TITLE", "GitHub MCP Server"), serverOpts)
+	ghServer := NewServer(cfg.Version, cfg.Translator("SERVER_NAME", "github"), cfg.Translator("SERVER_TITLE", "GitHub MCP Server"), serverOpts)
 
 	// Add middlewares. Order matters - for example, the error context middleware should be applied last so that it runs FIRST (closest to the handler) to ensure all errors are captured,
 	// and any middleware that needs to read or modify the context should be before it.
@@ -144,14 +144,20 @@ func addGitHubAPIErrorToContext(next mcp.MethodHandler) mcp.MethodHandler {
 
 // NewServer creates a new GitHub MCP server with the given version, server
 // name, display title, and options. If name or title are empty the defaults
-// "github-mcp-server" and "GitHub MCP Server" are used.
+// "github" and "GitHub MCP Server" are used.
+//
+// The "github" name aligns with the configuration key VS Code derives from
+// the human-readable title, so tool prefixes like "github/*" resolve. It
+// intentionally diverges from the MCP registry identifier
+// (io.github.github/github-mcp-server in server.json), which uses the source
+// repo's last segment per the registry naming convention.
 func NewServer(version, name, title string, opts *mcp.ServerOptions) *mcp.Server {
 	if opts == nil {
 		opts = &mcp.ServerOptions{}
 	}
 
 	if name == "" {
-		name = "github-mcp-server"
+		name = "github"
 	}
 	if title == "" {
 		title = "GitHub MCP Server"

pkg/github/server_test.go

@@ -207,7 +207,7 @@ func TestNewServer_NameAndTitleViaTranslation(t *testing.T) {
 		{
 			name:          "defaults when using NullTranslationHelper",
 			translator:    translations.NullTranslationHelper,
-			expectedName:  "github-mcp-server",
+			expectedName:  "github",
 			expectedTitle: "GitHub MCP Server",
 		},
 		{
@@ -242,7 +242,7 @@ func TestNewServer_NameAndTitleViaTranslation(t *testing.T) {
 		t.Run(tt.name, func(t *testing.T) {
 			t.Parallel()
 
-			srv := NewServer("v1.0.0", tt.translator("SERVER_NAME", "github-mcp-server"), tt.translator("SERVER_TITLE", "GitHub MCP Server"), nil)
+			srv := NewServer("v1.0.0", tt.translator("SERVER_NAME", "github"), tt.translator("SERVER_TITLE", "GitHub MCP Server"), nil)
 			require.NotNil(t, srv)
 
 			// Connect a client to retrieve the initialize result and verify ServerInfo.