university shuttle service

This is an aspect of public transport which is definitely undermapped. There was quite a detailed breakdown written a few years ago, which was never taken forward: https://wiki.openstreetmap.org/wiki/Proposed_features/Differentiation_for_routes_of_public_transport. This is probably the best place to start.

The only tag which seems to be in use is public_transport:type, which may be a little general (how many school only trains or tram services exist?).

'school=yes' is a poorly defined tag & I would avoid it, and definitely if it is for a university.

At least here (Nottingham) we have the following university associated services:

• Numbered stopping services between & within campuses & also the centre of the city. Provided by the University, and using regular public service bus stops.
• Shuttle services between large srudent acocmmodation sites (student villages) provided by the acoomodation provider.
• Long distance shuttles between the main campus & an outlying campus 15 kilometres away. Provided by the university. (These look closest to your example)

I'm also aware of universities which are slightly remote from railway stations have a dedicated shuttle for that purpose (Edge Hill University in Ormskirk), which might be a different type of service again.

Essentially these services are access=private, in the same way as workplace shuttle buses. I think suitable operator or brand tags are important (and some judicious choice may be needed if the university sub-contracts out the service, but brands it & provision is via a university card).