I am experiencing this same thing, 7 years later (!!) and can't find anything in the documentation or in the community about it.
Summary: Web Activity in MSI shows up for specific leads on the MSI widget on the lead record itself. But when I go to the MSI App/Tab (not associated with a specific Object), that same Lead does not show up in the Web Activity list.
I thought it had to do with Ownership of the Lead, but that doesnt appear to be the case, as my Sales VP can see web activity from records both owned by him and not owned by him, but this particular lead (and others) do not show up in his list.. I, as an admin, with the highest role, cannot see anything at all under Web Activity.
Anyone have an answer?