They missed my favourite form of urban transport the on road trams.
Clean, quite, non poluting, can penartrate town centres, can be given priority at junctions and reduce traffic including buses (which don't seem to mentioned). Althoughreducing traffic is contentious
The main study used data derived from censuses for 1991-2011, so the questions asked were census questions. There was a single choice for "public transport", and trams (and light rail) were listed as one mode within that only in 2001 and 2011. That's just one of those things about doing long-term research like this that you have to put up with - your choice of what to have asked people thirty years ago is a bit limited ...
I misread that table, which is rather ambiguous. From the study itself, it's clear that the census did offer a choice of individual modes, and even a write-in option for "other". So it was the study that aggregated them by only using the top-level choice (vehicle/public transport/walk/cycle). They did look at bus and train separately, and report that the beneficial association with health outcomes was much stronger for train than bus travel to work.
No doubt the aggregation was needed to get enough numbers in every box - this kind of study has boxes for a lot of combinations of possible confounding factors with travel mode, and statistics always needs quite big numbers if it's to work properly.
All the results are associations - not causal links. They say they looked at reverse causality (i.e. did poorer health determine travel mode), but I can't see the results of that in the text. But they do say that all of their attempts to adjust for known factors thought to be causal - most of which are to do with deprivation in some sense - will leave some common factors still present. Obviously it's hard to say how big that is - one of those "what do you know about what you don't know about" questions.
And of course the "no time machines" rule still applies.