Getting the pretty value of a Select field in Node twig template
When I call just this :

content.field_*machine_name*

  
  It display the pretty value but wrapped in a div that I don't want.


This is the correct way, you should if possible render a field in node twigs like this.

The div, that you don't want, is probably in the field twig, that gets invoked by using {{ content.field_*machine_name* }}.

You have to put a copy of field.html.twig in your theme and name it:

field--field_*machine_name*.html.twig

And look there for the div.