That's a light rail (tram) line, based on photos of it. Belongs in tier 4.
And if you find photos of it acting as an S-Bahn or as a U-Bahn, rather than the short street-running sections, it's not obviously dissimilar to other such systems. It's not as clear cut as you think.
Here's a train running an RB service in another part of Germany - it doesn't look much different to
the Karlsruhe light rail vehicles running on railways.
There are multiple distinguishing features
Nope. These are alleged distinguishing features. I note that you at least used 'typically' for most of them, acknowledging that these are more rule of thumb than hard-and-fast.
Street-running / right-of-way is somewhere where's the more complexity. I don't think anyone would think that London-Weymouth trains until the 80s would be light rail due to, having travelled over 100 miles (initially as intercity-style service),
the last mile being on-street to reach the port. There's quite a lot of this kind of thing for industrial usage (Trafford Park near Manchester was full of streets with train tracks running along them, but redevelopment has removed most), but it's not unheard of for passenger use. And there's also a difference between
street-running, where there's no horizontal segregation and so sharing of the right-of-way with other traffic, and
at-grade-running where there is a dedicated right-of-way alongside/in the middle of a street, with at-grade crossings of the ROW (not that dedicated ROW can't have at-grade crossings). Looking at Karlsruhe, only the Strassenbahn has the former, with the Stadtbahn only doing the latter.
High-floor vs low-floor is one of those geeky obsessions (different electrical systems are also similar) that don't mean much for function. To give an example from the UK, where railways are all high-floor (which isn't the case in, say, Germany or the USA) - the Manchester Metrolink, Croydon Tramlink and Midland Metro (to give them all their original names) are similar concept systems from similar eras - build some street-running route in the centre (in the Midland Metro's case, a short bit in Wolverhampton, until much later when they bothered with a Birmingham bit) and convert rail line(s) to be part of the tram network. Manchester kept the old rail routes' high platforms and has high-floor vehicles. The other two demolished the pre-existing platforms (or, occasionally, raised the track height) and put in low ones and have low-floor vehicles. The three networks are all functionally the same thing, with Manchester not being different because of the type of vehicles.
These distinctions also have some functional implications because light rail trains can generally handle steeper hills and navigate tighter curves than their heavy rail counterparts, and also have shorter stopping distances (they have to, to safely run in streets).
Perhaps, but the DLR is undeniably functionally Metro, despite having 'Light Railway' in its name and using stock designed to do all the things that you cite here as being a Light Rail speciality due to the nature of the system.
Watching videos about the newly opened REM in Montreal this lunchtime, there was a subtle question of whether it's designation as 'Light' makes sense. The answer was so obviously 'no' they didn't bother talking about it much.
"Light rail" is also admittedly an American term, but British English "tram" is used similarly.
Yes and no - the Karlsruhe Stadtbahn would be seen in British English as light rail as opposed to being tram, rather than the terms being equivalent. I think I might have treated it as S-bahn when allocating a tier, rather than U-bahn (per the Stadtbahn in the Rhine-Ruhr) due to the use of Sx line designations rather than Ux ones, and thus it is perhaps a tier too high, but it's not at the same tier as the regular tram network in the city (which is not Streetcar).
And the LA Metro Light Rail lines - like the DLR/REM, stuff the Green and Crenshaw lines that are completely segregated and grade-separated and only are light rail because they use the same vehicles as the other Light Rail lines. And those other ones, which are Light Rail, I'd find easier to put into 'Metro' than 'Tram' - they don't have street-running, but rather at-grade-running (so their own ROW), and the great majority of time there's vertical-separation as well as horizontal-segregation - the at-grade running is short.
