DNS Issue Again

How do you want it to resolve?
The issue might not be DNS but the server to which you are pointing this record where the issue is.
nslookup www.aspresolver.com.thissite.thisdomain.domain
or
dig www.aspresolver.com.thissite.thisdomain.domain.

telnet www.aspresolver.com.thissite.thisdomain.domain 80
GET http://www.aspresolver.com.thissite.thisdomain.domain/ HTTP/1.0

See what response you get.

not sure what you are trying to do so this is the best set of examples I can provide.

This resolved correctly externally:
Non-authoritative answer:
Name:    www.aspresolver.com.ezproxy.wscal.edu
Address:  174.77.164.85

Meaning that external DNS knows where it is at.  
When I put that address into my address bar, though, it goes into search mode instead of going to the page.

Additionally:
Z:\>ping www.aspresolver.com.ezproxy.wscal.edu

Pinging www.aspresolver.com.ezproxy.wscal.edu [192.168.100.57] with 32 bytes of
data:
Reply from 192.168.100.57: bytes=32 time<1ms TTL=128
Reply from 192.168.100.57: bytes=32 time<1ms TTL=128
Reply from 192.168.100.57: bytes=32 time<1ms TTL=128
Reply from 192.168.100.57: bytes=32 time<1ms TTL=128

Ping statistics for 192.168.100.57:
    Packets: Sent = 4, Received = 4, Lost = 0 (0% loss),
Approximate round trip times in milli-seconds:
    Minimum = 0ms, Maximum = 0ms, Average = 0ms

Would indicate that my internal DNS knows where to go.  I am not getting it to resolve in a web browser though.

Check your web server configuration and see whether the requests when they arrive to the 192.168.100.57 80 are the same as when they are hitting the external WAN IP port 80.

Are you forwarding the external connections to the same IP/POrt on the internal as the one in the example above?

Arnold, yes, the external IP is mapped to that internal IP.

Which port on the web server is the external IP pointing to versus the default web site your internal access is being directed to.

I would say that both are pointed at 443 since it is https.  How am I supposed to point it at a specific port for access?

could you double check your web server configuration?
Do you have only one default web site?

What is the web site running on?
IIS, Apache?
Apache
httpd -D DUMP_VHOSTS

IIS do you only have a single default web site? or do you have multiple sites.

Double check based on the log entries which site is being accessed when the traffic is going from the outside versus from the inside.

Arnold, this is a proxy server called EZ-Proxy.  As far as I know it is not running on either IIS or Apache.  There are no other sites on that computer.

The proxy is not the one serving the page. It has to resolve the hostname you want to access and has to be able to connect to wherever this URL is to retrieve the data.

Arnold,
I am not quite sure then on this answer.  Here is the link for EZ-Proxy initial set up:
http://www.oclc.org/support/services/ezproxy/documentation/download/install-win32.en.html

Arnold, when I went to test it again everything seemed to be working appropriately.