Edit: I realise after writing the answer that I sound rather negative. Sorry about that, but I can't phrase this in a positive fashion.
Answer: First, and foremost, you are mixing logic, with output. The template should not have logic in it, and should absolutely not care about why it was called.
When a user visits node/1, core will call node_load(1). This is in no way different from calling node_load(1) anywhere else.
A node template does also not execute as a direct result of loading a node. Templates are used when the theme layer renders a node, which is the completely other end of the response.
You can, in theory, call debug_backtrace() to get the call stack, and from there figure out more information about from where node_load was called, but there is almost certainly exactly 0 valid use-cases for this. I guarantee that you do not want to go down that path, and even considering it means that something is seriously wrong.
I have built Drupal-sites professionaly for years now, and have done some really, really ugly hacks during that time. Neither of them would come anywhere near analysing the stacktrace, in the theme layer.