How to see urls of incoming SQL Server traffic?
By the time the client connects, it typically doesn't present the hostname it used.  The client resolves the hostname in DNS and connects to the IP(s) that were returned from the DNS lookup.

Kerberos connections will fail if the client uses a hostname that doesn't correspond to an SPN registered for the SQL Server service account.  And NTLM can fail if the server is not configured to accept auth at the hostname the client uses.  But there's no general way to see this.  The configuration changes required to connect with these Auth schemes through a DNS alias are outlined here.

One thing you can do is to add an additional IP address to the server and configure the two DNS entries to resolve to different IP addresses.  That you would be able to monitor either at the network level or in sys.dm_exec_connections.