Not familiar with south western railroads, but links of the type of Toulouse-Narbonne-Carcassonne are only TER (transport express régional).
This is Occitanie's regional railroad network.
https://edito.igstatic.com/155791425...-Occitanie.JPG
Orange is TER lines operated by SNCF. These are the main regional lines widely funded by the national administration.
Green is secondary lines operated (and mostly funded, I guess) by the regional administration itself.
So EMarg and his people rode TER. This is nothing like HSR, guys. It doesn't feel the same at all. As a matter of fact, Toulouse itself is not served by any HSR yet.
It is the hub of the aviation industry in the country; so generally speaking, they don't fancy trains so much over there.