Internal network address contains my name - problematic?
What's your IP address? If it's an RFC 1918 address like 10.*, then you're being NATted to the Internet, and as a comorbid condition* your reverse DNS won't be visible to external sites. It will still allow the University to track you better internally, which is perhaps why they do it.
If it's not an RFC 1918 address, your DNS name likely does show up externally; you can check with a site like WhatIsMyIP (Reverse DNS Lookup).
*Having an RFC 1918 address implies that your reverse DNS won't be publicized on the Internet, because an RFC 1918 address must be translated (NAT, or Network Address Translation) before going out to the Internet. In almost all cases, NAT is one-to-many (one public address, many private addresses) or pool-to-many (some public addresses, many private addresses) so there is no one-to-one mapping between the internal RFC 1918 address and the external address the Internet sees traffic from. If you have hundreds or thousands of students being NATted to one or more public IP addresses, having that smaller number of public IP addresses be mapped to hundreds or thousands of names becomes meaningless. All of which I glibly summarized as "NAT means no external reverse DNS" above.
Whether this is visible to the outside world depends on what kind of firewall your university network uses.
You could try using the tools at www.whatismyip.com to see some of what's being exposed to the public Internet. The basic info at the home page will show you what public IP you have; you can compare this with the configuration of your PC to see if they're the same (the PC may have a private IP, and the public IP is is the address of the university's firewall).
If your internal IP is being exposed, the university may still be using split DNS to provide different name resolution internally and publicly. Use the Reverse DNS Lookup tool to see what name is being exposed to the public. I expect it will be a generic name derived automatically from the IP, not the one with your name in it.
Also, it's possible that the university uses dynamic address assignment that doesn't associate a specific IP with your PC permanently. Each time you connect to the network you may get a different IP. So if some outside site has your IP saved, and tries to look up the name later, they may get someone else's name because the assignment has changed. The "Last login" message you see in the login greeting on your server isn't affected by this because it saves the name when you login, not the IP.
Your hostname may be included in the Received headers added to any emails sent from that host (and not passing through NAT to the first MTA to handle them). You can find out if that's currently happening by sending a test email to an 'echo' reflector, but that doesn't tell you what server changes may affect that in the future.