I have a couple of final thoughts, and then unfortunately I'm out of ideas.
First, you can eliminate the firewall as an issue if you first try connecting using the browser on the same machine that the web client is being installed on. In fact, you should do you initial log-in from that machine until you get the web client working.
Second, you shouldn't have a different name for the machine when you connect from inside vs. outside of the network. The security certificate will have a specific name that it's expecting, so that is the name that you must use all of the time. If you aren't using a wildcard certificate, then you need to know the name that you supplied when you got the certificate. If you use this name: tspi-gpsvr01.tspi.com then that is the name that you should see defaulted in when you select the security certificate in the Web Client installer. Is that the name that you see? The name you supply in the installer must match the machine name that you use when you connect to the web client. That's not the name that you used in the URL way at the beginning of this thread, so that could be part of your issue.
Third, if you're sure the machine name in the security certificate matches the machine name that you supply in the installer, and that's the name that you're using when you connect to the machine, and things still don't work, you may have an issue with the security certificate. I'd check the purposes that are defined for the certificate to see what they are. (There's info about doing this in the web client install and admin guide.) Maybe an important one is missing.
Finally, I would go with the recommendation to try a self-signed certificate, just so you can verify that the other components of the web client are all functional. The Web Client Installation and Administration Guide will walk you through the entire process of using a self-signed security certificate.
I hope that one of these things helps you find your answer. Best of luck to you!