How did rotary converter commutators conduct electricity?
I think you may be misparsing the photo you linked — the commutator has solid copper sections separated by insulating material just like a motor commutator. (For example, if you look at the brush section marked "12" in the photo you can see the brushes reflected in the metal surface.) So, it is not different physically.

Now as to the number of segments. I'm not an expert in the field but I think I can explain this, so take this with a grain of salt:

On the AC side of things, the converter is like a synchronous AC motor — that is, the AC line frequency is a multiple of its rotational frequency. If the multiplication factor were 1 — the rotor has a single coil — then it would be spinning at 60 Hz, or in more conventional units for motors, 3600 RPM. We can clearly see that it is not going that fast — and if it was, it would be making more noise, putting more wear on the bearings and brushes, and experiencing more losses from air resistance.

Instead, the rotor and stator have many magnetic poles and correspondingly coils. Therefore, on the DC side, the polarity on the commutator also changes more than twice per full circle, so there are more than two brushes — but they are connected in two parallel sets. (If you used only two brushes, you would still get DC out, but you would not be evenly drawing from all of the rotor coils.)

I'm not sure, but I think in the edge-on view in the video starting at 1:00, you can see that the brushes are bolted alternatingly to two circular bus-bars.

I'm not sure that this is right — the Wikipedia article on rotary converters seems to indicate that there may be even more commutator segments than magnetic poles. If this is so, then the point of having many of them is to connect to the portion of the rotor coils which is experiencing the maximum EMF at any given instant, thus producing a smoother DC output (whereas, at the other extreme, having exactly two commutator poles would mean the output goes to zero but not below, like it does in a diode rectifier without any smoothing capacitor).