start button wrong label

Here's how this might be implemented:

// In your button build/render method:
$build['start_button'] = [
  '#type' => 'link',
  '#title' => $this->getUserSpecificLabel(),
  '#url' => $this->getUserSpecificUrl(),
  '#cache' => [
    'contexts' => ['user'], // This makes the button vary by user
  ],
];

// Or if this is a render array for the whole page:
$build['#cache']['contexts'][] = 'user';