Well, I've been inspecting the Print module code for a while and figured out the right solution. ( Mr. jdu's answer is still right too but if I have to build completely different templates for printer-friendly and default versions of nodes, it just won't be enough). So for those who may face this problem:
Print module creates a view mode called 'print'. When you get on /print/%nid page, a special function is fired that renders the node in 'print' view mode. Inspecting the code further we find that in 'print' view mode, in preprocess function, it adds the following theme_hook_suggestions:
if (($variables['elements']['#view_mode'] == 'print') && isset($variables['elements']['#print_format'])) {
$type = $variables['elements']['#node']->type;
$format = $variables['elements']['#print_format'];
$nid = $variables['elements']['#node']->nid;
$variables['theme_hook_suggestions'][] = "node__print";
$variables['theme_hook_suggestions'][] = "node__print__{$format}";
$variables['theme_hook_suggestions'][] = "node__print__{$format}__node__{$type}";
$variables['theme_hook_suggestions'][] = "node__print__{$format}__node__{$nid}";
}
Therefore, all we need is create, in my case, "node--print--html--node--nbt_text.tpl.php" (underscores from suggestions turn into dashes, my content type is 'nbt_text').