Add new existing field for existing content type
if I am not wrong, I believe, Site 1 is your local development environment, and site 2 is your production environment.

I wrote an update hook which will create the same config file (except order), when my module will be updated on other sites, where it was previously installed my module
Why would you write an update hook ? You can simply export the configs, and import them on prod.
If this does not suit you, I would suggest you to use Features module for this. It will make sure you get a working module which can be installed even without features module in place on Prod.