I have 2 different websites:
mysite.com
api.mysite.com
I want to be able to serve both of these websites from the same webserver (same IP) and same port number.
I have followed this document, and created the following websites:

This is how the binding for mysite.com looks like:

And this is how the binding for api.mysite.com looks like:

Note that the only way I could define 2 sites using the same IP and Port was to use a Host name for one of them.
With the above settings, mysite.com works but api.mysite.com does not work. If I stop mysite.com from IIS, then api.mysite.com starts working.
The document that I used as a guide explains that I need to use dnsmgmt.msc to define a CNAME for api.mysite.com, but here is where I am confused. I am using AWS and I have defined 2 CNAME records for these websites in Route 53. Do I still need to define a CNAME record for one of these websites in Windows?
If not, then how would I tell IIS that request containing api.mysite.com should be sent to website having the host name = api.mysite.com?
Update
The answer provided by Greg suggests that I would add mysite.com as Host name as well. If I follow this solution, how would I handle requests for www.mysite.com?
At the moment, any request for mysite.com is permanently redirected to www.mysite.com