The first link you've referenced is a folder, and within that folder is a bunch of services. This is a "Sample Server", as such, it's used to demonstrate different workflows and patterns. Grouped inside this folder is pretty much everything you can do to support Routing workflows.
I'm not real sure exactly what you're asking, but I can try to explain what's inside this service folder.
You see 3 different services types:
- Map Service
- Geoprocessing Service
- Network Analyst Service
Each of these services might be using the same data (well, it's highly likely they are), but each service does something different.
The map service is obviously for looking at the data on a map. This will be a feature class of streets and other data layers that provide context for your user.
The geoprocessing service allows a user to actually generate a route or other network function. There's actually 6 different network tools inside the GP Service. The user of the application would send a request to the service where inside network analyst tools run, and provide a result. The application then draws the result on the map.
The network analyst service works similar to the geoprocessing service, but it's used differently. The application consuming the NA service needs to understand how to work with this particular service, specifically how to send a request and handle the response. It's a whole other question/topic why and when you'd use a GP Service vs NA Service.
With the services explained, if you want to do something similar, it all starts in ArcMap or ArcGIS Pro with a streets featureclass. Publish your streets into a map service. Then create a Network Dataset and build/publish GP Services or publish the network into a NA Service. (This paragraph is an incredibly simplified explanation of the entire start to finish, somewhat complex workflow)