How to attach two domains to one site in IIS (for the local network and for the Internet)?

Question

IIS

There is software in the form of a site on IIS. The host is in the AD domain and, accordingly, for the local network, the site works according to the local domain name. It is necessary to make the host work from the Internet according to the domain name by which it appears on the Internet.
If you add bindings (bindings) and try to connect from the Internet, it simply redirects to the local name and, accordingly, access from the Internet ends. I tried to assign two IP addresses to one interface and separate the bindings by IP - it does not work. 404 issues. I tried to make two interfaces with different IPs on each and separate the bindings like this - it also does not work. I don't understand if it's possible to do what I need? And the second question is what is the redirect mechanism? why does it send all bindings to one name when it is explicitly said that there are two IPs with two different names.