I didn't want to be pushy of course! It was just in case as a new user you didn't know exactly how it works here! Take your time, and feel free not to "accept" my answer of course! :)

P.S. Since you are a new user and maybe you don't know exactly how it works here, may I say that once you get an answer which is satisfactory to you, then you should click to "accept" it.

Yes, I think the same.

You are welcome! Glad it was useful!

For your second question, why not? If you also have an infinitesimal version of it, you do have one more invariant to check!

Usually for bounded domains these are actual distances. But in general the constructions made to obtain these invariant objects merely produces pseudodistances. The point is they can be useful to distinguish domains (or more generally complex manifolds) even if they are not necessarily genuine distances. For example, you know that if for some domain your favourite invariant construction yields a true distance and for some other domain a pseudodistance which is not a distance, then you can directly conclude that these two domains are not biholomorphic!