While shopping for a leather jacket what test can I do in the store to
be sure that the material is truly genuine leather and not a very good
fake?

Smell it.

Might not a very good fake, also have a fake smell?

When I was in Korea, there were vendors who sold leather jackets. They
had a 'true test'.  Use a lighter.  A lighter will have no affect(
short term) on leather, it of course will start to melt any fake ,
almost immediately. Now, I don't propose you try this in a store, as
you may end up buying it, however, challenge the proprietor to the
test!