[client] Keep the device name on the context used to authenticate

Moving authentication onto the start context dropped the device name with it:
registration happens during login and reads the name from the context, so every
client registered under the host's hostname instead of its configured name. A
run of 100 embedded clients produced 191 peers all sharing one name.

Carry the value onto the start context explicitly so the deadline and the
device name travel together.

client/embed/embed.go

@@ -305,13 +305,18 @@ func (c *Client) Start(startCtx context.Context) error {
 	// never completes a stream leaves them retrying with no deadline of their
 	// own, so a caller's start timeout has to reach them. Neither outlives
 	// startup: authClient is closed below.
-	authClient, err := auth.NewAuth(startCtx, c.config.PrivateKey, c.config.ManagementURL, c.config)
+	// Carry the device name onto the start context: registration happens during
+	// login and reads it from there, so authenticating on a context without it
+	// would register every client under the host's hostname instead.
+	authCtx := context.WithValue(startCtx, system.DeviceNameCtxKey, c.deviceName) //nolint:staticcheck
+
+	authClient, err := auth.NewAuth(authCtx, c.config.PrivateKey, c.config.ManagementURL, c.config)
 	if err != nil {
 		return fmt.Errorf("create auth client: %w", err)
 	}
 	defer authClient.Close()
 
-	if err, _ := authClient.Login(startCtx, c.setupKey, c.jwtToken); err != nil {
+	if err, _ := authClient.Login(authCtx, c.setupKey, c.jwtToken); err != nil {
 		return fmt.Errorf("login: %w", err)
 	}
 	client := internal.NewConnectClient(ctx, c.config, c.recorder)