it seems like you are having a bit of a hard time getting what Frank Thomas is saying.
Assuming you are at host A. If you want to get to host D you need to go through both routers B and C. B and C, both have interfaces on the public network (22.214.171.124 and 132.277.62.1 respectively). However, their internal routing network is allocated by private space. That is, to route from B to C you have to go over a private network.
If you were to traceroute from A to D you would see something like this:
traceroute to 126.96.36.199 (188.8.131.52), 30 hops max, 60 byte packets
1 184.108.40.206 1.103 ms 1.107 ms 1.097 ms
2 10.10.10.2 1.535 ms 1.625 ms 2.172 ms
3 132.277.62.122 6.891 ms 6.937 ms 6.927 ms
Please note the times are irrelevant for this example.
So even though both sides of the network are public, router B has a route to D via C. Now, please understand - in the real world B probably also has a route which goes to the Internet. However, in this case D's network had a better route (or in the technical parlance - a lower metric) on B via C. Therefore, you see the private network.
Why this happens depends on the specific network. I will hazard a guess. You seem to be on a University network. I say this based on the fact that your internal IP addresses had a DNS entry. Which have a
univ in them. If this is indeed the case, you are now routing within a University - or intra-University. As a result, the best route from one University to another may in fact be an internal route instead of going out over the Internet. They may have a circuit that they ran themselves, or they might be using MPLS or Frame Relay to create a private circuit.
Hope that helps.