So here is what I think is happening. The move 9...Bb4 is obviously not good because White can just take the undefended bishop. However the fact that Nc6 was played previously now supports the bishop move to b4, making this more likely to be a good move. Leela uses the previously played moves from the history planes to determine which squares have been recently supported, and makes moves there more likely by increasing the policy.
To check whether this is correct, I examined how Leela evaluated the Bb4 policy before and after 9...a5 10.d5 which also supported Bb4. With history, Bb4 policy = 18.4%. Without history, Bb4 policy = 0.74%. Making another move that does not support b4 does not improve the policy (e.g. 9...Ng6 10.d5 gives Bb4 policy of 0.06% with history and 0.12% without). These results are consistent with my theory.
I'm sure this isn't the only use that Leela makes of the history planes, but I haven't seen this possibility mentioned before.
if you can find that, then all it takes is to study samples of the networks as they developed over time, for which data already exists.
Does Leela play these openings differently because of different learning experiences ?
history provides trajectories (amongst other things). if your main line shows a theme of making progress then good. History.
This is the best "I set up such-and-such position with Leela and look what happened" post of all time, I think. It gave me insight into why Leela would use history planes, when I've been pretty solidly of the "the position is the position" school of the debate, as limited as it's been as Chris already mentioned.